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\begin{document}


\begin{center}
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\mbox{\null}
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\begin{center}
\vskip 1cm{\LARGE\bf Ramanujan Type Trigonometric Formulas:\\
\vskip .1in
The General Form for the Argument $\displaystyle\frac{2\, \pi}{7}$}
\vskip 1cm
\large
Roman Witu{\l}a \\
Institute of Mathematics \\
Silesian University of Technology \\
Kaszubska 23 \\
Gliwice 44-100 \\
Poland \\
\href{mailto:roman.witula@polsl.pl}{\tt roman.witula@polsl.pl} \\
\end{center}


\vskip .2in

\begin{abstract}
In this paper,
we present many general identities connected with the classical
Ramanujan equality.
Moreover, we give Binet formulas for an accelerator sequence for
Catalan's constant.
\end{abstract}

\vskip .2in

\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{defin}[theorem]{Definition}
\newenvironment{definition}{\begin{defin}\normalfont\quad}{\end{defin}}
\newtheorem{examp}[theorem]{Example}
\newenvironment{example}{\begin{examp}\normalfont\quad}{\end{examp}}
\newtheorem{rema}[theorem]{Remark}
\newenvironment{remark}{\begin{rema}\normalfont\quad}{\end{rema}}


\section{Introduction}

The main objective of this paper is to obtain some general trigonometric formulas
related to the known Ramanujan equality~\cite{Berndt,Berndt2,Berndt3,Kreczmar,Shevelev1988}:
\begin{equation}\label{w-1.0}
\sqrt[3]{\cos  \alpha} +
\sqrt[3]{\cos  2\alpha} +
\sqrt[3]{\cos  4\alpha} =
\sqrt[3]{\tfrac{1}{2} \big( 5 - 3\, \sqrt[3]{7}  \big)},
\end{equation}
where $\alpha=2 \pi/7$.
Other formulas of this type, referring to the ninth and eleventh primitive roots of unity,
etc., will be published in separate papers.
The present paper, in a way, is a~continuation
of previous papers~\cite{WitulaSlota-A7,WitulaSlota-A7s,WitulaSlota-A11}
and I will take advantage of some results from those papers.
The quasi-Fibonacci sequences of the seventh order discussed in the above mentioned papers
are applied here for describing some attractive formulas involving radicals.

The paper is divided into five parts:
\begin{list}{}{}
\item[--] Section 2~-- where some striking
equalities related to equality~(\ref{w-1.0}) are presented.
Furthermore, Binet formulas for two new sequences
$\{\mathcal{S}_{n}\}$ and $\{\mathcal{S}_{n}^{*}\}$
are derived,
which, at the same time, resolves the problem of an algebraic description
of the zeros of polynomials $x^3-\sqrt[3]{7}x-1$ and $x^3-\sqrt[3]{49}x-1$
(see Remark~\ref{rem2-1}).

\item[--] Section 3~-- where the fundamental formula~(\ref{ram-gw}) for
a~sum of the cube roots of the three roots of a cubic polynomial is given.

\item[--] Section 4~-- where many basic sequences of integers,
reals and complex numbers, introduced and discussed earlier by the authors
in papers~\cite{WitulaSlota-A7,WitulaSlota-A7s,WitulaSlota-A11}, are presented.
In addition, some new relations between the elements of the sequences are discussed.
Furthermore, a~new description of Binet's formula
is introduced for an accelerator sequence for Catalan's constant,
which, naturally, makes it possible to extend this formula for
all integers (see Remark~\ref{rem4-4}).

\item[--] Section 5~-- where applications of the formula~(\ref{ram-gw})
to many special kinds of polynomial of degree three are given.
This section contains many
Ramanujan type trigonometric formulas.
Moreover in Remark~\ref{rem4.3} the nontrivial theoretical discussion
of some  numerical case is presented.
\end{list}


\section{Delicious}

Now we are going to prove the following three interesting identities:
\begin{multline}\label{w-nr2}
\sqrt[3]{\frac{\cos \alpha}{\cos 2\alpha}}\,
\big( 2\, \cos \alpha \big)^{k}
+
\sqrt[3]{\frac{\cos 2\alpha}{\cos 4\alpha}}\,
\big( 2\, \cos 2\alpha \big)^{k}
%+ {}\\
{}+
\sqrt[3]{\frac{\cos 4\alpha}{\cos \alpha}}\,
\big( 2\, \cos 4\alpha \big)^{k}
={}\\
{}=
\sqrt[3]{\frac{\cos \alpha}{\cos 2\alpha}}\,
\big( 2\, \cos 2\alpha \big)^{k+1}
+
\sqrt[3]{\frac{\cos 2\alpha}{\cos 4\alpha}}\,
\big( 2\, \cos 4\alpha \big)^{k+1}
+ {}\\
{}+
\sqrt[3]{\frac{\cos 4\alpha}{\cos \alpha}}\,
\big( 2\, \cos \alpha \big)^{k+1}
%={}\\
{}=
\sqrt[3]{7}\, \psi_{k},
\end{multline}
where $ \psi_{0}=-1$, $ \psi_{1}=0$, $ \psi_{2}=-3$ and
$$
 \psi_{k+3} +  \psi_{k+2} -2\,  \psi_{k+1} -  \psi_{k} =0,
\qquad k\in \mathbb{Z};
$$
%
\begin{multline}\label{w-nr1}
\sqrt[3]{\frac{\cos \alpha}{\cos 4\alpha}}\,
\big( 2\, \cos \alpha \big)^{k}
+
\sqrt[3]{\frac{\cos 2\alpha}{\cos \alpha}}\,
\big( 2\, \cos 2\alpha \big)^{k}
+
\sqrt[3]{\frac{\cos 4\alpha}{\cos 2\alpha}}\,
\big( 2\, \cos 4\alpha \big)^{k}
={}\\
{}=
\sqrt[3]{\frac{\cos 2\alpha}{\cos \alpha}}\,
\big( 2\, \cos \alpha \big)^{k+1}
+
\sqrt[3]{\frac{\cos 4\alpha}{\cos 2\alpha}}\,
\big( 2\, \cos 2\alpha \big)^{k+1}
+ {}\\
{}+
\sqrt[3]{\frac{\cos \alpha}{\cos 4\alpha}}\,
\big( 2\, \cos 4\alpha \big)^{k+1}
{}=
\sqrt[3]{49}\, \varphi_{k},
\end{multline}
where $ \varphi_{0}=0$, $ \varphi_{1}=-1$,
$ \varphi_{2}=1$ and
$$
 \varphi_{k+3} +  \varphi_{k+2} -2\,  \varphi_{k+1} -  \varphi_{k} =0,
\qquad k\in \mathbb{Z};
$$
\begin{multline}\label{w-nr3}
\sqrt[3]{\sec 2\alpha}\,
\big( 2\, \cos \alpha \big)^{k}
+
\sqrt[3]{\sec 4\alpha}\,
\big( 2\, \cos 2\alpha \big)^{k}
+
\sqrt[3]{\sec \alpha}\,
\big( 2\, \cos 4\alpha \big)^{k}
={}\\
{}=
\delta_k\, \sqrt[3]{8-6\, \sqrt[3]{7}} +
\sigma_k\, \sqrt[3]{6\, \big( 1+\sqrt[3]{7}\big)^2} +
\xi_k\, \sqrt[3]{2\, \big( 5-3\,\sqrt[3]{7}\big)^2}
={}\\
{}=
\sqrt[3]{
f_{3k+1} + 6
- \frac{3}{\sqrt[3]{2}} \,
\Big(
\sqrt[3]{ \mathcal{S}_{3k+1,8} + \sqrt{\mathcal{T}_{3k+1,8}} }
+
\sqrt[3]{ \mathcal{S}_{3k+1,8} - \sqrt{\mathcal{T}_{3k+1,8}} }\,\,
\Big)},
\end{multline}
where
\begin{align*}
\delta_0 &= 1, & \delta_1 &= 0, & \delta_2 &= 0,\\
\sigma_0 &= 0, & \sigma_1 &= -1, & \sigma_2 &= 0,\\
\xi_0 &= 0, & \xi_1 &= 0, & \xi_2 &= 1,
\end{align*}
$$
\mathbb{X}_{k+3} +  \mathbb{X}_{k+2} -2\,  \mathbb{X}_{k+1} -  \mathbb{X}_{k} =0,
\qquad k=0,1,2,\ldots,
$$
for every $\mathbb{X}\in \{\delta,\sigma,\xi\}$,
whereas the sequences $f_{3k+1}$, $\mathcal{S}_{3k+1,8}$ and $\mathcal{T}_{3k+1,8}$
are defined by formulas~(\ref{f2s-w3g}), (\ref{w6-e}) and~(\ref{w6-f})
(in Section~\ref{roz3}, other sequences occurring in the definition
of sequences $\mathcal{S}_{3k+1,8}$ and $\mathcal{T}_{3k+1,8}$ are defined as well).
The first twelve values of numbers $\psi_n$ and $\varphi_n$ are presented
in Table~\ref{tabela}.

Moreover an interesting numerical link to the formula~(\ref{w-nr2})
are the considerations from Remark~\ref{rem4.3}.

\medskip\noindent
\textit{Proof of formulas~(\ref{w-nr2})--(\ref{w-nr3}).}

For $k=0,1,2$, the formulas (\ref{w-nr2})--(\ref{w-nr3})
follow from~(\ref{sf-w2}), (\ref{sf-w5}), (\ref{sf-w7a}),
(\ref{sf-w8}) and (\ref{sf-w11}), and (or) from the following equalities
(in both cases, equality~(\ref{w-0}) below for $\mathbb{X}=0$ is needed):
\begin{multline*}
\sqrt[3]{\frac{\cos\alpha}{\cos 2\alpha}}\,
\big( 2\, \cos\alpha \big)^2 +
\sqrt[3]{\frac{\cos 2\alpha}{\cos 4\alpha}}\,
\big( 2\, \cos 2\alpha \big)^2 +
\sqrt[3]{\frac{\cos 4\alpha}{\cos \alpha}}\,
\big( 2\, \cos 4\alpha \big)^2  = {}\\
{}=
\sqrt[3]{\frac{\cos\alpha}{\cos 2\alpha}}\,
\big( 2+2\, \cos 2\alpha \big)
{}+
\sqrt[3]{\frac{\cos 2\alpha}{\cos 4\alpha}}\,
\big( 2+2\, \cos 4\alpha \big) +
\sqrt[3]{\frac{\cos 4\alpha}{\cos\alpha}}\,
\big( 2+2\, \cos\alpha \big) ={}\\
%\end{multline*}
%\begin{multline*}
{}=
2\, \sqrt[3]{7} \, \psi_0 +
2\, \Big(
\sqrt[3]{\cos \alpha\, \cos^2 2\alpha} +
\sqrt[3]{\cos 2\alpha\, \cos^2 4\alpha} +
\sqrt[3]{\cos 4\alpha\, \cos^2 \alpha}\,
\Big) = {}\\
{}=
-2\, \sqrt[3]{7} +
\sqrt[3]{\frac{\cos 2\alpha}{\cos 4\alpha}} +
\sqrt[3]{\frac{\cos 4\alpha}{\cos \alpha}} +
\sqrt[3]{\frac{\cos \alpha}{\cos 2\alpha}}  =
-2\, \sqrt[3]{7} + \sqrt[3]{7} \, \psi_0 = -3\, \sqrt[3]{7},
\end{multline*}
and
\begin{multline*}%equation*}
\sqrt[3]{\frac{\cos\alpha}{\cos 4\alpha}}\,
\big( 2\, \cos \alpha \big)^2 +
\sqrt[3]{\frac{\cos 2\alpha}{\cos \alpha}}\,
\big( 2\, \cos 2\alpha \big)^2 +
\sqrt[3]{\frac{\cos 4\alpha}{\cos 2\alpha}}\,
\big( 2\, \cos 4\alpha \big)^2  ={}\\
{}=
\sqrt[3]{\frac{\cos \alpha}{\cos 4\alpha}}\,
\big( 2 + 2\, \cos 2\alpha \big) +
\sqrt[3]{\frac{\cos 2\alpha}{\cos \alpha}}\,
\big( 2 + 2\, \cos 4\alpha \big) +
\sqrt[3]{\frac{\cos 4\alpha}{\cos 2\alpha}}\,
\big( 2 + 2\, \cos \alpha \big) ={}%\\
\end{multline*}%equation*}
%\pagebreak
\begin{multline*}
{}=2\, \sqrt[3]{49}\, \varphi_0 +
\left(\sqrt[3]{\frac{\cos 2\alpha}{\cos 4\alpha}}\right)^2+
\left(\sqrt[3]{\frac{\cos 4\alpha}{\cos \alpha}}\right)^2+
\left(\sqrt[3]{\frac{\cos \alpha}{\cos 2\alpha}}\right)^2
={}\\
=\!\!
\left(\sqrt[3]{\frac{\cos 2\alpha}{\cos 4\alpha}} +
\sqrt[3]{\frac{\cos 4\alpha}{\cos \alpha}}+
\sqrt[3]{\frac{\cos \alpha}{\cos 2\alpha}}\right)^{\!\!2}
-2\!
\left(
\sqrt[3]{\frac{\cos 2\alpha}{\cos \alpha}} +
\sqrt[3]{\frac{\cos \alpha}{\cos 4\alpha}}+
\sqrt[3]{\frac{\cos 4\alpha}{\cos 2\alpha}}
\right)
\!\!=\\
{}=
\Big( \sqrt[3]{7}\, \psi_0 \Big)^2
-2\,\sqrt[3]{49}\, \varphi_0=\sqrt[3]{49}.
\end{multline*}
Now let us set
$$
\mathfrak{B}_n := \sum_{k=0}^{2} x_k\, \big( \cos \big(2^{k}\alpha\big) \big)^n,
$$
where $x_k\in \mathbb{R}$, $k=1,2,3$, are given.
Then, from Newton's formula we obtain
$$
\mathfrak{B}_{n+3}+\mathfrak{B}_{n+2}-2\,\mathfrak{B}_{n+1} -\mathfrak{B}_{n} =0
$$
since~ \cite{Kreczmar,WitulaSlota-A7,WitulaSlota-A7s}:
\begin{equation}\label{w-0}
\prod_{k=0}^{2}
\Big( \mathbb{X} - 2\, \cos \big(2^{k}\alpha \big) \Big)
= \mathbb{X}^3 + \mathbb{X}^2 -2\, \mathbb{X} -1.
\end{equation}
Hence, on account of the definitions of sequences $\varphi_k$ and $\psi_k$,
$k\in \mathbb{N}$, by applied induction arguments
the formulas~(\ref{w-nr2}) and~(\ref{w-nr1}) follow.
Similarly, by applying ~(\ref{sf-w1}), (\ref{sf-w4}) and~(\ref{sf-w7})
we deduce the first part of~(\ref{w-nr3}).
The second part of~(\ref{w-nr3}) from~(\ref{w6-d}) follows.




\begin{remark}\label{rem2-1}
Let $a,b,c\in \mathbb{C}$ and $a+b+c=0$.
Put
$$
s_k := a^k + b^k + c^k,
\qquad k\in \mathbb{N}.
$$
Then the following relations hold \cite{Kreczmar,Modenov}:
\begin{align*}
2\, s_4 &= s_{2}^{2}; & 6\, s_5 &=5\, s_2\, s_3; \\
6\, s_7 &= 7\, s_3\, s_4; & 10\, s_7 &= 7\, s_2\, s_5;\\
25\, s_3\, s_7 &= 21 s_{5}^{2}; & 50\, s_{7}^{2} &= 49\, s_4\, s_{5}^{2}
\end{align*}
and the respective Newton formula has the form
$$
s_{n+3} = a\, b\, c\, s_n + \frac{1}{2}\, s_2\, s_{n+1},
\qquad n\in \mathbb{N}.
$$
Hence, and from~(\ref{w-nr1}) for $k=0$, we get
$$
\mathcal{S}_{n+3} =  \sqrt[3]{7}\, \mathcal{S}_{n+1} + \mathcal{S}_n ,
$$
where $\mathcal{S}_0=3$, $\mathcal{S}_1=0$, $\mathcal{S}_2=2\,\sqrt[3]{7}$,
$$
\mathcal{S}_n :=
\left(\frac{\cos \alpha}{\cos 4\alpha}\right)^{\!n/3}+
\left(\frac{\cos 2\alpha}{\cos \alpha}\right)^{\!n/3}+
\left(\frac{\cos 4\alpha}{\cos 2\alpha}\right)^{\!n/3}
$$
which implies the following formula
$$
\mathcal{S}_n = \widehat{a}_n+ \widehat{b}_n\, \sqrt[3]{7} + \widehat{c}_n\, \sqrt[3]{49},
$$
where
\begin{align*}
\widehat{a}_{n+3} &= \widehat{a}_n +7\, \widehat{c}_n,\\
\widehat{b}_{n+3} &= \widehat{b}_n + \widehat{a}_{n+1},\\
\widehat{c}_{n+3} &= \widehat{c}_n + \widehat{b}_{n+1},\\
\widehat{a}_0 &=3, \quad \widehat{a}_1=0,\quad \widehat{a}_2=0,\\
\widehat{b}_0 &=0, \quad \widehat{b}_1=0,\quad \widehat{b}_2=1,\\
\widehat{c}_0 &=0, \quad \widehat{c}_1=0,\quad \widehat{c}_2=0.
\end{align*}
On the other hand, from~(\ref{w-nr2}) for $k=1$, we obtain
$$
\mathcal{S}_{n+3}^{*} =  \frac{1}{2}\,
\mathcal{S}_{2}^{*}\, \mathcal{S}_{n+1}^{*} + \mathcal{S}_{n}^{*} ,
\qquad n\in \mathbb{N}_0,
$$
where $\mathcal{S}_{0}^{*}=3$, $\mathcal{S}_{1}^{*}=0$,
\begin{equation*}
%\begin{multline*}
\mathcal{S}_{n}^{*} :=
\bigg(2\,\cos \alpha\,
\sqrt[3]{\frac{\cos \alpha}{\cos 2\alpha}}\, \bigg)^{\!\!n}+
\bigg(2\,\cos 2\alpha\,
\sqrt[3]{\frac{\cos 2\alpha}{\cos 4\alpha}}\, \bigg)^{\!\!n}
%+{}\\
%{}
+
\bigg(2\,\cos 4\alpha\,
\sqrt[3]{\frac{\cos 4\alpha}{\cos \alpha}}\, \bigg)^{\!\!n},
\quad n\in \mathbb{N},
%\end{multline*}
\end{equation*}
and by~(\ref{w-0}) for $\mathbb{X}=0$:
\begin{multline*}
\mathcal{S}_{2}^{*} :=
\left(
\frac{\cos \alpha}{\cos 2\alpha}
\right)^{\!\!2/3}
\big( 2+2\,\cos 2\alpha\big) +
\left(
\frac{\cos 2\alpha}{\cos 4\alpha}
\right)^{\!\!2/3}\!\!
\big( 2+2\,\cos 4\alpha\big) %+{}\\
{}+
\left(
\frac{\cos 4\alpha}{\cos \alpha}
\right)^{\!\!2/3}\!\!
\big( 2+2\,\cos \alpha \big) ={}\\
{}=
2\, \left(
\left(\frac{\cos \alpha}{\cos 2\alpha}
\right)^{\!\!1/3} +
\left(\frac{\cos 2\alpha}{\cos 4\alpha}
\right)^{\!\!1/3} +
\left(\frac{\cos 4\alpha}{\cos \alpha}
\right)^{\!\!1/3}\,
\right)^{\!\!2}
-{}\\
{}
-4\,
\left(
\left(\frac{\cos \alpha}{\cos 4\alpha}
\right)^{\!\!1/3} +
\left(\frac{\cos 4\alpha}{\cos 2\alpha}
\right)^{\!\!1/3} +
\left(\frac{\cos 2\alpha}{\cos \alpha}
\right)^{\!\!1/3}\,
\right)
+{}\\
{}+
\big( 2\, \cos \alpha \big)^{1/3}\,
\big( 4\, \cos \alpha\, \cos 2\alpha \big)^{1/3} +
\big( 2\, \cos 2\alpha \Big)^{1/3}\,
\big( 4\, \cos 2\alpha\, \cos 4\alpha \big)^{1/3} +{}\\
{}+
\big( 2\, \cos 4\alpha \big)^{1/3}\,
\big( 4\, \cos \alpha\, \cos 4\alpha \big)^{1/3}
=
2\, (\sqrt[3]{7}\,\psi_0)^2 - 4\, \mathcal{S}_1 + \mathcal{S}_1 = 2\, \sqrt[3]{49}.
\end{multline*}
So, we have
$$
\mathcal{S}_{n+3}^{*} = \sqrt[3]{49}\, \mathcal{S}_{n+1}^{*} + \mathcal{S}_{n}^{*},
\qquad n\in \mathbb{N}_0,
$$
which implies
$$
\mathcal{S}_{n}^{*} = a_{n}^{*}+ b_{n}^{*}\, \sqrt[3]{7} + c_{n}^{*}\, \sqrt[3]{49},
$$
where
\begin{align*}
{a}^{*}_{n+3} &= {a}^{*}_n + 7\, {b}^{*}_{n+1},\\
{b}^{*}_{n+3} &= {b}^{*}_n + 7\, {c}^{*}_{n+1},\\
{c}^{*}_{n+3} &= {c}^{*}_n + {a}^{*}_{n+1},%\\
\end{align*}
\begin{align*}
{a}^{*}_0 &=3, \quad {a}^{*}_1=0,\quad {a}^{*}_2=0,\\
{b}^{*}_0 &=0, \quad {b}^{*}_1=0,\quad {b}^{*}_2=0,\\
{c}^{*}_0 &=0, \quad {c}^{*}_1=0,\quad {c}^{*}_2=2.
\end{align*}
We note that the elements
$\mathcal{S}_{3n}=\widehat{a}_{n}$ and
$\mathcal{S}_{3n}^{*}={a}^{*}_{n}$, $n=0,1,\ldots$, are all integers.
\end{remark}





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\section{Some theoretical deliberations}\label{roz3a}



Let us assume that $\xi_{1}$, $\xi_{2}$, $\xi_{3}$
are complex roots of the following polynomial with complex coefficients
$$
f(z) := z^{3} + p\,z^{2} + q\,z + r.
$$
The symbols $\sqrt[3]{\xi_{1}}$, $\sqrt[3]{\xi_{2}}$, $\sqrt[3]{\xi_{3}}$
will denote any of the third complex roots of the numbers
$\xi_{1}$, $\xi_{2}$ and $\xi_{3}$, respectively
(only in the case where $\xi_{1}$, $\xi_{2}$ and $\xi_{3}$ are real numbers,
we will assume that $\sqrt[3]{\xi_{1}}$, $\sqrt[3]{\xi_{2}}$ and $\sqrt[3]{\xi_{3}}$
also denote the respective real roots).



Let us assume that
$$
A :=
\left(
\sqrt[3]{\xi_{1}} + \sqrt[3]{\xi_{2}} + \sqrt[3]{\xi_{3}}
\right)^{3}
$$
and
$$
B := \left( \sqrt[3]{\xi_{1}}\, \sqrt[3]{\xi_{2}} +
\sqrt[3]{\xi_{1}}\, \sqrt[3]{\xi_{3}} +
\sqrt[3]{\xi_{2}}\, \sqrt[3]{\xi_{3}}  \right)^{3}.
$$
Thus, the numbers
$$
\sqrt[3]{\xi_{1}} + \sqrt[3]{\xi_{2}} + \sqrt[3]{\xi_{3}}
\qquad  \mbox{ and } \qquad
\sqrt[3]{\xi_{1}}\, \sqrt[3]{\xi_{2}} +
\sqrt[3]{\xi_{1}}\, \sqrt[3]{\xi_{3}} +
\sqrt[3]{\xi_{2}}\, \sqrt[3]{\xi_{3}}
$$
belong to the sets of the third complex roots of
$A$ and~$B$, respectively, which, for conciseness of notation, will be denoted 
by the symbols
$\sqrt[3]{A}$ and~$\sqrt[3]{B}$, respectively. In other words, we have
$$
\sqrt[3]{\xi_{1}} + \sqrt[3]{\xi_{2}} + \sqrt[3]{\xi_{3}} \in \sqrt[3]{A}
$$
and
$$
\sqrt[3]{\xi_{1}}\, \sqrt[3]{\xi_{2}} +
\sqrt[3]{\xi_{1}}\, \sqrt[3]{\xi_{3}} +
\sqrt[3]{\xi_{2}}\, \sqrt[3]{\xi_{3}} \in \sqrt[3]{B} .
$$
Hence, after two-sided raising of the numbers to the third power,
we obtain the following formulas:
$$
A = \xi_{1} + \xi_{2} + \xi_{3} +
3\, \sum _{k \neq l} \left( \sqrt[3]{\xi_{k}} \right)^{2}\, \sqrt[3]{\xi_{l}} +
6\, \sqrt[3]{\xi_{1}}\, \sqrt[3]{\xi_{2}}\, \sqrt[3]{\xi_{3}} ,
$$
and
$$
B = \xi_{1}\, \xi_{2} + \xi_{1}\, \xi_{3} + \xi_{2}\, \xi_{3} +
3\! \sum _{k \neq l} \left( \sqrt[3]{\xi_{k}} \right)^{2}\! \sqrt[3]{\xi_{l}}\,
\sqrt[3]{\xi_{1}}\, \sqrt[3]{\xi_{2}}\, \sqrt[3]{\xi_{3} }+
6\! \left( \sqrt[3]{\xi_{1}}\, \sqrt[3]{\xi_{2}}\, \sqrt[3]{\xi_{3}} \right)^{2},
$$
where
$$
\sqrt[3]{\xi_{1}}\, \sqrt[3]{\xi_{2}}\, \sqrt[3]{\xi_{3}} \in
\sqrt[3]{\xi_{1}\, \xi_{2} \,\xi_{3}} = \sqrt[3]{-r} = - \sqrt[3]{r}.
$$
Also here, for abbreviation,
the product $({-}1)\, \sqrt[3]{\xi_{1}}\,\sqrt[3]{\xi_{2}}\, \sqrt[3]{\xi_{3}}$
will be denoted by the symbol $\sqrt[3]{r}$.

Taking into account Vi\`ete's formulas for polynomial  $f(z)$, the expressions
for $A$ and~$B$ can be attributed the following form (from now on, symbols
$\sqrt[3]{A}$ and~$\sqrt[3]{B}$ will mean the properly selected elements
from sets $\sqrt[3]{A}$ and~$\sqrt[3]{B}$, respectively):
\begin{equation}\label{ram-w1}
A = -p + 3\, \sqrt[3]{A}\, \sqrt[3]{B} + 3\sqrt[3]{r}
\end{equation}
and
\begin{equation}\label{ram-w2}
B = q - 3\, \sqrt[3]{A}\, \sqrt[3]{B}\, \sqrt[3]{r} - 3\, \left( \sqrt[3]{r} \right) ^{2} .
\end{equation}

By multiplying the first of these equations by $\sqrt[3]{r}$
and adding the equations side-by-side, we obtain
\begin{equation}\label{ram-w3}
B = q - (A + p)\, \sqrt[3]{r} .
\end{equation}

At the same time, the equation~(\ref{ram-w1}) yields
$$
3\, \sqrt[3]{A}\, \sqrt[3]{B} = A + p - 3\, \sqrt[3]{r} ,
$$
i.e.,
$$
27\, A\, B = \left( A + p - 3\, \sqrt[3]{r} \right) ^{3} ,
$$
hence, with respect to~(\ref{ram-w3}) we obtain
\begin{multline*}
27\, A\, \big( q - (A + p)\, \sqrt[3]{r} \big) = A^{3} + p^{3} - 27\, r + {}\\
{}+
3\, \big( A^{2}\, (p - 3\sqrt[3]{r}) + A\,  (p^{2} + 9\, (\sqrt[3]{r})^{2} ) -
3\, p^{2}\, \sqrt[3]{r} + 9\, p\, (\sqrt[3]{r})^{2} \big) - 18\, A\, p\, \sqrt[3]{r},
\end{multline*}
and having rearrange the summands (with respect to the powers of~$A$),
we obtain the basic equality
\begin{equation}\label{ram-w4}
A^{3} + 3\, \big( p + 6\, \sqrt[3]{r} \big) A^{2} +
3\, \big( p^{2} + 3\, p\, \sqrt[3]{r} + 9\, (\sqrt[3]{r})^{2} - 9\, q \big) A +
\big( p - 3\, \sqrt[3]{r} \big)^{3} = 0.
\end{equation}

By applying Cardano's formula to this polynomial, we get the following basic formula
(the right side of the formula below means a~properly selected
third root of the number present in the formula):
\begin{multline}\label{ram-gw}
\sqrt[3]{A}=\sqrt[3]{\xi_1} +\sqrt[3]{\xi_2}+\sqrt[3]{\xi_3} = {}\\
{}=
\sqrt[3]{
-p-6\, \sqrt[3]{r} - \frac{3}{\sqrt[3]{2}}\,
\Big(
\sqrt[3]{\mathcal{S}+\sqrt{\mathcal{T}}} +
\sqrt[3]{\mathcal{S}-\sqrt{\mathcal{T}}}\,
\Big)
},
\end{multline}
where
\begin{align*}
\mathcal{S} &:= p\, q +6\, q\, \sqrt[3]{r} +6\, p\, \sqrt[3]{r^2} +9\, r,\\
\mathcal{T} &:= p^2\, q^2 -4\, q^3 - 4\, p^3\, r + 18\, p\,q\, r -27\, r^2.
\end{align*}
In the case when $\mathcal{T}\geq 0$, $\mathcal{S}\in \mathbb{R}$, $r\in \mathbb{R}$,
we assume that all the roots appearing here are real.

\begin{remark}\label{ram-rem1}
We note that, if in the formula~(\ref{ram-w4}) the following condition holds
$$
\big( p + 6\, \sqrt[3]{r} \big)^{2} = p^{2} + 3\, p\, \sqrt[3]{r} +
9\, (\sqrt[3]{r})^{2} - 9\, q ,
$$
i.e.,
\begin{equation}\label{ram-w5}
p\, \sqrt[3]{r} + 3\, (\sqrt[3]{r})^{2} + q = 0,
\end{equation}
then the equation~(\ref{ram-w4}) could be given in the form
$$
\big( A + p + 6\, \sqrt[3]{r} \big)^{3} = \big( p + 6\, \sqrt[3]{r} \big)^{3}
- \big( p - 3\, \sqrt[3]{r} \big) ^{3} ,
$$
hence we get
\begin{equation}\label{ram-w6}
A = - p - 6\, \sqrt[3]{r} + \sqrt[3]{ \big( p + 6\, \sqrt[3]{r} \big)^{3}
- \big( p - 3\, \sqrt[3]{r} \big)^{3} }.
\end{equation}
\end{remark}


\begin{remark}
The analysis which enabled describing the value of~$A$ by means of
coefficients of polynomial~$f(z)$ comes from the 
papers~\cite{Shevelev1988,Kreczmar}
(see also~\cite{GrzymkowskiWitula}).
\end{remark}

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\section{Basic sequences}\label{roz3}

We will now provide definitions of a~dozen basic sequences
(not only integer sequences) used further in the paper.
For more information concerning these sequences
(including the trigonometric relationships defining their terms),
see the papers~\cite{WitulaSlota-A7,WitulaSlota-A7s}.

The sequences $\{A_{n}(\delta)\}_{n=0}^{\infty}$,
$\{B_{n}(\delta)\}_{n=0}^{\infty}$ and $\{C_{n}(\delta)\}_{n=0}^{\infty}$
are the so-called quasi-Fibonacci numbers of the seventh order
described in~\cite{WitulaSlota-A7} by means of relations
\begin{equation}
(1+\delta\, (\xi^{k}+\xi^{6k}))^n =
A_n(\delta)+B_n(\delta)\, (\xi^{k}+\xi^{6k})+C_n(\delta)\, (\xi^{2k}+\xi^{5k})
\end{equation}
for $k=1,2,3$,
where $\xi\in\mathbb{C}$
is a~primitive root of unity of the seventh order
($\xi^7=1$ and $\xi\neq 1$), $\delta\in \mathbb{C}$, $\delta\neq 0$.
These sequences satisfy the following recurrence relations
\begin{equation}
\left\{\begin{array}{l}
A_0(\delta )=1,\ \  B_0(\delta )=C_0(\delta )=0, \\
A_{n+1}(\delta )=A_n(\delta )+2\,\delta\, B_n(\delta )-\delta\, C_n(\delta ), \\
B_{n+1}(\delta )=\delta\, A_n(\delta )+B_n(\delta ), \\
C_{n+1}(\delta )=\delta\, B_n(\delta )+(1-\delta )\,C_n(\delta ),
\end{array}\right.
\label{aa}
\end{equation}
for every $n\in\mathbb{N}$.

Two auxiliary sequences  $\{\mathcal{A}_{n}(\delta)\}_{n=0}^{\infty}$
and $\{\mathcal{B}_{n}(\delta)\}_{n=0}^{\infty}$
connected with these ones are defined by the following relations:
\begin{equation}\label{w-and}
\mathcal{A}_{n}(\delta) := 3\, A_{n}(\delta) - B_{n}(\delta)
-C_{n}(\delta)
\end{equation}
and
\begin{equation}
\mathcal{B}_{n}(\delta) := \frac{1}{2} \big( \big(\mathcal{A}_{n}(\delta)\big)^{2} -
\mathcal{A}_{2n}(\delta) \big).\label{w-bnd}
\end{equation}
Furthermore, to simplify notation, we will write
\begin{equation}\label{nr100}
\mathcal{A}_{n}=\mathcal{A}_{n}(1),\
\mathcal{B}_{n}=\mathcal{B}_{n}(1),\
A_{n}=A_{n}(1),\
B_{n}=B_{n}(1)\ \mbox{and}\
C_{n}=C_{n}(1),
\end{equation}
for every $n\in \mathbb{N}$.


We note that the elements of the sequences $\{\mathcal{A}_{n}\}_{n=0}^{\infty}$,
$\{{A}_{n}\}_{n=0}^{\infty}$,
$\{{B}_{n}\}_{n=0}^{\infty}$ and
$\{{C}_{n}\}_{n=0}^{\infty}$
respectively, satisfy the following recurrence relation
(see~\cite[eq.~(3.20)]{WitulaSlota-A7s}):
$$
\mathbb{X}_{n+3} -2\, \mathbb{X}_{n+2} -\mathbb{X}_{n+1} + \mathbb{X}_n \equiv 0.
$$
Simultaneously, the elements of sequence  $\{\mathcal{B}_{n}\}_{n=0}^{\infty}$
by~\cite[eqs.~(3.18), (3.13)]{WitulaSlota-A7s} satisfy the following relation
\begin{equation}\label{w-Nr-0}
\mathcal{B}_{n+3} + \mathcal{B}_{n+2} -2\, \mathcal{B}_{n+1} - \mathcal{B}_n \equiv 0.
\end{equation}

\begin{remark}
The sequence  $\{\mathcal{B}_{n}\}_{n=0}^{\infty}$
is an accelerator sequence for Catalan's constant
(see~\cite[sequence \seqnum{A094648}]{Sloane}
and papers~\cite{WitulaSlota-A7,WitulaSlota-A7s}).
\end{remark}

The elements of the sequences
$\{a_n\}_{n=0}^{\infty}$,
$\{b_n\}_{n=0}^{\infty}$ and
$\{c_n\}_{n=0}^{\infty}$ are defined by the following recurrence relations:
$$
a_{0}=b_{0}=c_{0}=\sqrt{7}
$$
and
\begin{equation}\label{f2s-w1g}
\left\{
\begin{array}{l}
a_{n+1} = 2\, a_{n} + b_{n},\\
b_{n+1} = a_{n} + 2\, b_{n} - c_{n},\\
c_{n+1} = c_{n} - b_{n},\\
\end{array}
\right.
\end{equation}
for $n=0,1,2,\ldots$.

Moreover, we will use the following sequences
\begin{equation}\label{f2s-w2ga}
\left\{
\begin{array}{l}
\overline{\alpha}_{n} := c_{n+1},\\
\overline{\beta}_{n}  := {-}a_{n} - b_{n} ,\\
\overline{\gamma}_{n} := a_{n},\\
\end{array}
\right.
\end{equation}

Next, sequences
$\{f_n\}_{n=0}^{\infty}$,
$\{g_n\}_{n=0}^{\infty}$,
$\{h_n\}_{n=0}^{\infty}$
and
$\{H_n\}_{n=0}^{\infty}$
are defined in the following way
$$
f_0=g_0=h_0=-1,
$$
and
\begin{equation}\label{f2s-w3g}
\left\{
\begin{array}{ll}
f_{n+1} = f_{n} + g_{n},& \quad n \geq 0,\\
g_{n+1} = f_{n} + h_{n},& \quad n \geq 0,\\
h_{n} = \mathcal{B}_{n+1},& \quad n \geq 1,\\
2\, H_n = f_{n}^{2} +f_{2n} - g_{n}^{2} + h_{n}^{2} - h_{2n+1}
-2\, h_{2n}, & \quad n \geq 0,\\
\end{array}
\right.
\end{equation}
(the numbers $ \mathcal{B}_{n}$ are defined by the formula~(\ref{nr100}) above).

And at last, the elements of sequences
$\{u_n\}_{n=0}^{\infty}$, $\{v_n\}_{n=0}^{\infty}$, $\{w_n\}_{n=0}^{\infty}$,
$\{x_n\}_{n=0}^{\infty}$, $\{y_n\}_{n=0}^{\infty}$ and
$\{z_n\}_{n=0}^{\infty}$ are defined by
\begin{equation}
\left\{
\begin{array}{l}
u_{n+1} = x_{n},\\
v_{n+1} = -y_{n}-z_{n} = x_{n} - \sqrt{7} \, z_{n-1},\\
w_{n+1} = y_{n}-x_{n},\\
x_{n+1} = u_{n} - w_{n},\\
y_{n+1} = w_{n} - v_{n},\\
z_{n+1} = 2\, z_{n-1} - v_{n},
\end{array}
\right.
\end{equation}
for $n=0,1,2,\ldots$, where
$u_0=v_0=w_0=-1$, $x_0=y_0=z_0=\sqrt{7}$ and $z_1=7$.


\begin{remark}
All the recurrence sequences above have a~third order; the selective identities,
Binet formulas and generating functions, and some different identities for
these numbers, are presented in papers \cite{WitulaSlota-A7,WitulaSlota-A7s}.
For example, the following equivalent recurrence relations hold
$$
\mbox{(\ref{f2s-w1g})}\ \ \Longleftrightarrow\ \
\mathbb{X}_{n+3} - 5\, \mathbb{X}_{n+2} + 6\, \mathbb{X}_{n+1} - \mathbb{X}_{n} = 0,
$$
for $n=0,1,2,\ldots$, and
$\mathbb{X}\in \{a,b,c\}$, and
$a_{0}=b_{0}=c_{0}=\sqrt{7}$,
$a_{1}=3\,\sqrt{7}$, $b_{1}=2\,\sqrt{7}$, $c_{1}=0$,
$a_{2}=8\,\sqrt{7}$, $b_{2}=\sqrt{7^3}$, $c_{1}=-2\, \sqrt{7}$.
We also have
\begin{align*}
a_{n} &= 2^{2n+1}\, \Big[
\sin \alpha\,
\big(\! \cos 4\alpha \big)^{\!2n} +
\sin 2\alpha\,
\big(\! \cos \alpha \big)^{\!2n} +
\sin 4\alpha\,
\big(\! \cos 2\alpha \big)^{\!2n} \Big],\\
b_{n} &= 2^{2n+1}\, \Big[
\sin 2\alpha\,
\big(\! \cos 4\alpha \big)^{\!2n} +
\sin 4\alpha\,
\big(\! \cos \alpha \big)^{\!2n} +
\sin \alpha\,
\big(\! \cos 2\alpha \big)^{\!2n} \Big],\\
c_{n} &= 2^{2n+1}\, \Big[
\sin 4\alpha\,
\big(\! \cos 4\alpha \big)^{\!2n} +
\sin \alpha\,
\big(\! \cos \alpha \big)^{\!2n} +
\sin 2\alpha\,
\big(\! \cos 2\alpha \big)^{\!2n} \Big],
\end{align*}
etc.
\end{remark}

\begin{remark}
Now we present some new identities for the above sequences
(identities~(\ref{nr-alpha})--(\ref{nr-delta}), (\ref{w3.16})--(\ref{w3.19})),
which will be used in subsection~\ref{poroz4.2}.
These identities significantly complete those obtained in paper~\cite{WitulaSlota-A7s}.
By~\cite[Lemma~3.14 (a)]{WitulaSlota-A7s}, equality~(\ref{w-0})
and \cite[eq.~(3.21)]{WitulaSlota-A7s}, the following identity holds
\begin{equation}\label{nr-alpha}
(-1)^{n}\, \mathcal{A}_{n} = H_{n+1}.
\end{equation}
Hence, by~\cite[eq.~(3.23)]{WitulaSlota-A7s},  we obtain
\begin{multline}\label{nr-beta}
2\, \cos \alpha\,
\big( 2\, \cos 2\alpha \big)^{-n} +
2\, \cos 2\alpha\,
\big( 2\, \cos 4\alpha \big)^{-n} +
2\, \cos 4\alpha\,
\big( 2\, \cos \alpha \big)^{-n}
={}\\
{}=
(-1)^{n}\, \big( \mathcal{A}_{n} + \mathcal{A}_{n-1} - \mathcal{A}_{n-2} \big)
{}=
(-1)^{n}\, \big( \mathcal{A}_{n+1} - \mathcal{A}_{n} \big),
\end{multline}
and next, by~\cite[eq.~(3.22)]{WitulaSlota-A7s}, we obtain
\begin{multline}\label{nr-gamma}
2\, \cos \alpha\,
\big( 2\, \cos 4\alpha \big)^{-n} +
2\, \cos 2\alpha\,
\big( 2\, \cos \alpha \big)^{-n} +
2\, \cos 4\alpha\,
\big( 2\, \cos 2\alpha \big)^{-n}
={}\\
{}=
(-1)^{n}\, \big( \mathcal{A}_{n+1} - \mathcal{A}_{n-1} + \mathcal{A}_{n-2} - 7\, A_{n} \big)
{}=
(-1)^{n}\, \big( \mathcal{A}_{n-1} - \mathcal{A}_{n+1} \big),
\end{multline}
since, by~\cite[Remark~3.11]{WitulaSlota-A7s}, we have the identity
$$
\mathcal{A}_{n+3} =
2\, \mathcal{A}_{n+2} + \mathcal{A}_{n+1} - \mathcal{A}_{n}
$$
and by~\cite[Remark~3.8]{WitulaSlota-A7s}, we have
$$
4\, \mathcal{A}_{n} - \mathcal{A}_{n-2} = 7\, {A}_{n}.
$$
Moreover, the following formula can be easily generated
\begin{equation}\label{nr-delta}
\mathcal{B}_{n} = g_{n+1} + h_{n+1}.
\end{equation}
By~\cite[eq.~(3.18)]{WitulaSlota-A7s}, we have
\begin{equation}\label{w3.15}
h_{n-1}=\mathcal{B}_{n},
\end{equation}
which by~(\ref{nr-delta}) implies
\begin{equation}\label{w3.16}
g_{n+1}=\mathcal{B}_{n}-\mathcal{B}_{n+2},
\end{equation}
and next, by~\cite[eq.~(3.12)]{WitulaSlota-A7s} and by~(\ref{w-Nr-0}), we get
\begin{equation}\label{w3.17}
f_{n}=g_{n+1}-h_{n}=\mathcal{B}_{n}-\mathcal{B}_{n+2}-\mathcal{B}_{n+1}=
-\mathcal{B}_{n-1}-\mathcal{B}_{n}.
\end{equation}
Moreover, we obtain
\begin{equation}\label{w3.18}
f_{n}+g_{n}=-\mathcal{B}_{n}-\mathcal{B}_{n+1}
\end{equation}
and
\begin{equation}\label{w3.19}
f_{n}+h_{n}=\mathcal{B}_{n}-\mathcal{B}_{n+2}.
\end{equation}
\end{remark}

\begin{remark}\label{rem4-4}
From~(\ref{w3.15}) we get (see~\cite[eq.~(3.11)]{WitulaSlota-A7s}):
$$
\mathcal{B}_{n} = 2^n\,
\big( \cos^n\alpha + \cos^n 2\alpha + \cos^n 4\alpha \big),
$$
from which the next form can be deduced
\begin{equation}\label{nnnnn}
\mathcal{B}_{n} =
\Big( \frac{\sin 2\alpha}{\sin  \alpha} \Big)^{\! n} +
\Big( \frac{\sin 4\alpha}{\sin 2\alpha} \Big)^{\! n} +
\Big( \frac{\sin  \alpha}{\sin 4\alpha} \Big)^{\! n}.
\end{equation}
This forms of Binet's formula for $\mathcal{B}_{n}$
are more attractive than the Sloane's ones (see~\cite[sequence \seqnum{A094648}]{Sloane}).
Moreover, the formula~(\ref{nnnnn}) makes it possible to extend the definition $\mathcal{B}_{n}$
for negative integers~$n$.
So $\{\mathcal{B}_{n}\}_{n=-\infty}^{\infty}$ is a~two-sided sequence
of integers which is defined for all integers either by recurrence formula~(\ref{w-Nr-0}),
or equivalently by Binet's formula~(\ref{nnnnn}).
\end{remark}


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\section{Applications of the formula~(\ref{ram-gw})}




\subsection{Some special formulas}


\subsubsection*{By~\cite[eq.~(3.30)]{WitulaSlota-A7s}}

\noindent
for $n=1$:
\begin{equation}\label{sf-w1}
\sqrt[3]{\sec \alpha} +
\sqrt[3]{\sec 2\alpha} +
\sqrt[3]{\sec 4\alpha} =
\sqrt[3]{8-6\,\sqrt[3]{7}}\, ;
\end{equation}
%\medskip

\noindent
for $n=2$:
\begin{equation}\label{sf-w2}
\sqrt[3]{\frac{\cos \alpha}{\cos 2\alpha}}+
\sqrt[3]{\frac{\cos 2\alpha}{\cos 4\alpha}}+
\sqrt[3]{\frac{\cos 4\alpha}{\cos \alpha}}=
-\sqrt[3]{7}\, ;
\end{equation}
%\medskip

\noindent
for $n=3$:
\begin{equation}\label{sf-w3}
\cos 2\alpha\,
\sqrt[3]{2\, \cos \alpha}
+
\cos 4\alpha\,
\sqrt[3]{2\, \cos 2\alpha}
+
\cos \alpha\,
\sqrt[3]{2\, \cos 4\alpha} %={}\\
{}=
\frac{1}{2}\, \sqrt[3]{5 + 3\, \sqrt[3]{7} - 3\, \sqrt[3]{49}}
\, ;
\end{equation}
%\medskip


\noindent
for $n=4$:
\begin{equation}\label{sf-w4}
\cos 2\alpha\,
\sqrt[3]{\sec 4\alpha}
+
\cos 4\alpha\,
\sqrt[3]{\sec \alpha}
+
\cos \alpha\,
\sqrt[3]{\sec 2\alpha} =
-\sqrt[3]{\frac{3}{4}\, \big( 1 + \sqrt[3]{7} \big)^{2}
}
\, ;
\end{equation}
%\medskip


\noindent
for $n=5$:
\begin{equation}\label{sf-w5}
\cos 2\alpha\,
\sqrt[3]{
\frac{\cos 2\alpha}{
\cos 4\alpha}
}
+
\cos 4\alpha\,
\sqrt[3]{
\frac{\cos 4\alpha}{
\cos \alpha}
}
+
\cos \alpha\,
\sqrt[3]{
\frac{\cos \alpha}{
\cos 2\alpha}
}
=
0
\, ;
\end{equation}
%\medskip

\noindent
for $n=6$:
\begin{multline}\label{sf-w6}
\cos^2 (2\alpha) \,
\sqrt[3]{2\, \cos \alpha}
+
\cos^2 (4\alpha) \,
\sqrt[3]{2\, \cos 2\alpha}
+
\cos^2 (\alpha) \,
\sqrt[3]{2\, \cos 4\alpha} ={}\\
{}=
\frac{1}{4}\, \sqrt[3]{12\, \sqrt[3]{7} - \big( 4 + 3\, \sqrt[3]{7}\,\big)^2 }
\, ;
\end{multline}
%\medskip


\noindent
for $n=7$:
\begin{equation}\label{sf-w7}
 \cos^2 (2\alpha) \,
\sqrt[3]{\sec 4\alpha}
+
\cos^2 (4\alpha) \,
\sqrt[3]{\sec \alpha}
+
\cos^2 (\alpha)\,
\sqrt[3]{\sec 2\alpha} %={}\\
{}=
\frac{1}{4}\, \sqrt[3]{2\,\big( 5 - 3\, \sqrt[3]{7}\,\big)^2 }
\, ;
\end{equation}
%\medskip


\noindent
for $n=8$:
\begin{multline}\label{sf-w7a}
\cos^3 (2\alpha)\,
\sqrt[3]{
\frac{\cos \alpha}{
\cos 2\alpha}
}
+
\cos^3 (4\alpha)\,
\sqrt[3]{
\frac{\cos 2\alpha}{
\cos 4\alpha}
}
+
\cos^3 (\alpha)\,
\sqrt[3]{
\frac{\cos 4\alpha}{
\cos \alpha}
}
={}\\
{}=
\frac{1}{2}\,
\bigg(\,
\cos^2 (2\alpha)\,
\sqrt[3]{
\frac{\cos 2\alpha}{
\cos 4\alpha}
}
+
\cos^2 (4\alpha)\,
\sqrt[3]{
\frac{\cos 4\alpha}{
\cos \alpha}
}
+
\cos^2 (\alpha)\,
\sqrt[3]{
\frac{\cos \alpha}{
\cos 2\alpha}
}\,
\bigg)
=
-\frac{3}{8}\, \sqrt[3]{7}\, ,
\end{multline}
which also can be generated from ~(\ref{sf-w2}).
\medskip


\begin{remark}
The formula~(\ref{sf-w7}) also follows
from~(\ref{w-1.0}) and~(\ref{w-0}) for $\mathbb{X}=0$:
\begin{multline*}
2\, \cos^2 (2\alpha) \,
\sqrt[3]{\sec 4\alpha}
+
2\, \cos^2 (4\alpha) \,
\sqrt[3]{\sec \alpha}
+
2\, \cos^2 (\alpha)\,
\sqrt[3]{\sec 2\alpha} ={}\\
{}=
\sum_{k=0}^{2} \sqrt[3]{\sec ( 2^{k}\alpha )} +
\sum_{k=0}^{2} \Big( \sqrt[3]{\cos ( 2^{k}\alpha )}\, \Big)^2 = {}\\
{}=
\sum_{k=0}^{2} \sqrt[3]{\sec ( 2^{k}\alpha )} +
\Big( \sum_{k=0}^{2} \sqrt[3]{\cos ( 2^{k}\alpha )}\, \Big)^2
-2\,
\sum_{k=0}^{2} \sqrt[3]{\cos ( 2^{k}\alpha )\,
\cos ( 2^{k+1}\alpha )}
= {}\\
{}=
\Big( \sum_{k=0}^{2} \sqrt[3]{\cos ( 2^{k}\alpha )}\, \Big)^2.
\end{multline*}
\end{remark}


\subsubsection*{By~\cite[eq.~(3.31)]{WitulaSlota-A7s}}


\noindent
for $n=2$:
\begin{equation}\label{sf-w8}
\sqrt[3]{
\frac{\cos 2\alpha}{
\cos \alpha}
}
+
\sqrt[3]{
\frac{\cos 4\alpha}{
\cos 2\alpha}
}
+
\sqrt[3]{
\frac{\cos \alpha}{
\cos 4\alpha}
}
=
0
\, ;
\end{equation}
%\medskip

\noindent
for $n=3$:
\begin{equation}\label{sf-w9}
\cos \alpha\,
\sqrt[3]{\cos 2\alpha}
+
\cos 2\alpha\,
\sqrt[3]{\cos 4\alpha}
+
\cos 4\alpha\,
\sqrt[3]{\cos \alpha}
=
-\frac{1}{2}\, \sqrt[3]{1 + \frac{3}{2}\, \sqrt[3]{49}}
\, ;
\end{equation}
%\medskip

\noindent
for $n=4$:
\begin{equation}\label{sf-w10}
\cos \alpha\,
\sqrt[3]{\sec 4\alpha}
+
\cos 4\alpha\,
\sqrt[3]{\sec 2\alpha}
+
\cos 2\alpha\,
\sqrt[3]{\sec \alpha} =
\sqrt[3]{\frac{9}{4}\, \big( 2 - \sqrt[3]{7} \big) }
\, ;
\end{equation}
%\medskip


\noindent
for $n=5$:
\begin{equation}\label{sf-w11}
\cos^2 (2\alpha)\,
\sqrt[3]{
\frac{\cos 4\alpha}{
\cos 2\alpha}
}
+
\cos^2 (4\alpha)\,
\sqrt[3]{
\frac{\cos \alpha}{
\cos 4\alpha}
}
+
\cos^2 (\alpha)\,
\sqrt[3]{
\frac{\cos 2\alpha}{
\cos \alpha}
}
=
-\frac{1}{4}\, \sqrt[3]{49}
;
\end{equation}
%\medskip

\noindent
for $n=6$:
\begin{multline}\label{sf-w12}
\cos^2 (2\alpha) \,
\sqrt[3]{\cos 4\alpha}
+
\cos^2 (\alpha) \,
\sqrt[3]{\cos 2\alpha}
+
\cos^2 (4\alpha) \,
\sqrt[3]{\cos \alpha} ={}\\
{}=
2^{-7/3}\, \sqrt[3]{47 + 3 \, \sqrt[3]{7} - 12\, \sqrt[3]{49} }
\, ;
\end{multline}
%\medskip


\noindent
for $n=7$:
\begin{multline}\label{sf-w13}
\cos^2 (\alpha) \,
\sqrt[3]{\sec 4\alpha}
+
\cos^2 (4\alpha) \,
\sqrt[3]{\sec 2\alpha}
+
\cos^2 (2\alpha) \,
\sqrt[3]{\sec \alpha} ={}\\
{}=
-2^{5/3}\, \sqrt[3]{ 73 + 36\, \sqrt[3]{7} + 3\, \sqrt[3]{49} }
\, ;
\end{multline}
%\medskip


\begin{remark}
We note that
\begin{multline*}
2\, \cos^2 (2\alpha) \,
\sqrt[3]{2\, \cos \alpha}
+
2\, \cos^2 (4\alpha) \,
\sqrt[3]{2\, \cos 2\alpha}
+
2\, \cos^2 (\alpha) \,
\sqrt[3]{2\, \cos 4\alpha}
={}\\
{}=
\Big(
\sqrt[3]{2\, \cos \alpha}+
\sqrt[3]{2\, \cos 2\alpha}+
\sqrt[3]{2\, \cos 4\alpha}
\Big) + {}\\
{}+
\cos 4\alpha \,
\sqrt[3]{2\, \cos \alpha} +
\cos \alpha \,
\sqrt[3]{2\, \cos 2\alpha} +
\cos 2\alpha \,
\sqrt[3]{2\, \cos 4\alpha} ={}\\
{}\stackrel{(\ref{w-1.0}), (\ref{sf-w9})}{=}
\sqrt[3]{5-3\, \sqrt[3]{7}} - \frac{1}{2}\, \sqrt[3]{2+3\, \sqrt[3]{49}}
\stackrel{(\ref{sf-w6})}{=}
\frac{1}{2}\, \sqrt[3]{12\, \sqrt[3]{7} - \big( 4 + 3\, \sqrt[3]{7}\big)^{2}\, },
\end{multline*}
which implies the identity
$$
\sqrt[3]{16 + 12\, \sqrt[3]{7} + 9\, \sqrt[3]{49}\, } =
2\, \sqrt[3]{3\, \sqrt[3]{7}-5} + \sqrt[3]{2+3\, \sqrt[3]{49}}.
$$
\end{remark}




\subsubsection*{By~\cite[eq.~(3.32)]{WitulaSlota-A7s}}

\noindent
for $n=2$:
\begin{equation}\label{sf-w14}
\sqrt[3]{
\frac{\cos \alpha}{
\cos^2 (2\alpha)}
}
+
\sqrt[3]{
\frac{\cos 2\alpha}{
\cos^2 (4\alpha)}
}
+
\sqrt[3]{
\frac{\cos 4\alpha}{
\cos^2 (\alpha)}
}
=
\sqrt[3]{
2\, \big( 11 - 3\, \sqrt[3]{49} \big)
}
\, ;
\end{equation}
%\medskip

\noindent
for $n=3$:
\begin{equation}\label{sf-w15}
\frac{\sqrt[3]{\cos \alpha}}{
\cos 2\alpha}
+
\frac{\sqrt[3]{\cos 2\alpha}}{
\cos 4\alpha}
+
\frac{\sqrt[3]{\cos 4\alpha}}{
\cos \alpha}
=
-\sqrt[3]{
36\, \big(1 +\sqrt[3]{7} \big)
}
\, ;
\end{equation}
%\medskip




\subsubsection*{By~\cite[eq.~(3.33)]{WitulaSlota-A7s}}

\noindent
for $n=2$:
\begin{equation}\label{sf-w16}
\sqrt[3]{
\frac{\cos \alpha}{
\cos^2 (4\alpha)}
}
+
\sqrt[3]{
\frac{\cos 2\alpha}{
\cos^2 (\alpha)}
}
+
\sqrt[3]{
\frac{\cos 4\alpha}{
\cos^2 (2\alpha)}
}
%= {}\\
{}=
\sqrt[3]{
6\, \big(-1 + \sqrt[3]{7} - \sqrt[3]{49}\big)
}
=
-2\, \sqrt[3]{\frac{6}{1+\sqrt[3]{7}}}
\, ;
\end{equation}
%\medskip

\noindent
for $n=3$:
\begin{equation}\label{sf-w17}
\frac{\sqrt[3]{\cos \alpha}}{
\cos 4\alpha}
+
\frac{\sqrt[3]{\cos 2\alpha}}{
\cos \alpha}
+
\frac{\sqrt[3]{\cos 4\alpha}}{
\cos 2\alpha}
=
\sqrt[3]{
4\, \big(26 -6\, \sqrt[3]{7} - 3\, \sqrt[3]{49}\big)
}
\, ;
\end{equation}
%\medskip



\subsubsection*{By~\cite[eq.~(3.34)]{WitulaSlota-A7s}}

\noindent
for $n=2$ we get~(\ref{sf-w1});
\medskip

\noindent
for $n=3$:
\begin{equation}\label{sf-w19}
\big( 2\, \cos \alpha \big)^{-2/3} +
\big( 2\, \cos 2\alpha \big)^{-2/3} +
\big( 2\, \cos 4\alpha \big)^{-2/3} %= {}\\
{}=
\sqrt[3]{
12 + 6\, \sqrt[3]{7} + 3\, \sqrt[3]{49}
}\, ;
\end{equation}
%\medskip










\subsubsection*{By~\cite[eq.~(4.30)]{WitulaSlota-A7s}}



\noindent
for $n=1$:
\begin{multline}\label{sf-w20}
\sqrt[18]{7} \, \Big( \sqrt[3]{\cot \alpha} +
\sqrt[3]{\cot 2\alpha} +
\sqrt[3]{\cot 4\alpha}\, \Big)
={}\\
=
\sqrt[3]{ \sqrt[3]{49}  -6 + 3\,
\sqrt[3]{3\, (1-\sqrt[3]{7}+\sqrt[3]{49})}
-3\, \sqrt[3]{5+3\,\sqrt[3]{7}-3\,\sqrt[3]{49}}\,
}
\, ;
\end{multline}
%\medskip

\noindent
for $n=2$:
\begin{multline}\label{sf-w22}
\sqrt[9]{\frac{7}{8}} \, \Big(
\sqrt[3]{\cot \alpha\, \csc \alpha } +
\sqrt[3]{\cot 2\alpha\, \csc 2\alpha} +
\sqrt[3]{\cot 4\alpha\, \csc 4\alpha}\,
\Big)
={}\\
=
\sqrt[3]{6 - 2\, \sqrt[3]{7}  -
3\, \sqrt[3]{3\, (1+\sqrt[3]{7})^2} -
3\, \sqrt[3]{-26+6\,\sqrt[3]{7}+3\,\sqrt[3]{49}}
}
\, ,
\end{multline}
i.e,
\begin{equation}\label{sf-w22a}
\sum_{k=0}^{2}
\sqrt[3]{\cot 2^k\alpha\, \csc 2^k\alpha } %= {}\\
{}=
\sqrt[3]{ \frac{6}{\sqrt[3]{7}}\,
\Big( 2 -
\frac{2}{3}\, \sqrt[3]{7} -
\sqrt[3]{3(1+\sqrt[3]{7})^2} -
\sqrt[3]{3(1+\sqrt[3]{7})^2-27}\,
\Big)}
\, .
\end{equation}

%\medskip

\noindent
Moreover from~(\ref{gwiazdka}) below we have for $n=1$:
\begin{multline}\label{sf-w21}
\sqrt[3]{\tan \alpha} +
\sqrt[3]{\tan 2\alpha} +
\sqrt[3]{\tan 4\alpha}
={}\\
=
\sqrt[18]{7} \,
\sqrt[3]{ 3\, \sqrt[3]{3\, (1-\sqrt[3]{7}+\sqrt[3]{49})}
-3\, \sqrt[3]{5+3\,\sqrt[3]{7}-3\,\sqrt[3]{49}}
- 6 - \sqrt[3]{7}
}
\, ;
\end{multline}
%\medskip

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\subsection{Some general formulas}\label{poroz4.2}


\textbf{1.}
By~\cite[eq.~(2.1)]{WitulaSlota-A7s} we obtain (for $n=0,1,2,\ldots$):
\begin{multline}\label{ram-2-w1}
\big( 2\, \cos \alpha \big)^{n/3} +
\big( 2\, \cos 2\alpha \big)^{n/3} +
\big( 2\, \cos 4\alpha \big)^{n/3} = {}\\
{}=
\sqrt[3]{
\mathcal{B}_{n} + 6
- \frac{3}{\sqrt[3]{2}} \,
\Big(
\sqrt[3]{ \mathcal{S}_{n,1} + \sqrt{\mathcal{T}_{n,1}} }
+
\sqrt[3]{ \mathcal{S}_{n,1} - \sqrt{\mathcal{T}_{n,1}} }\,\,
\Big)
},
\end{multline}
where
\begin{align}
\mathcal{S}_{n,1} &:=
(-1)^{n-1}\, \mathcal{A}_{n}\, \big( \mathcal{B}_{n} + 6 \big)
- 6\,\mathcal{B}_{n} -9,\label{ram-2-w2}\\
\mathcal{T}_{n,1} &:=
\mathcal{A}_{n}^{2}\,\mathcal{B}_{n}^{2} +
4\, (-1)^{n-1}\, \mathcal{A}_{n}^{3} - 4\,\mathcal{B}_{n}^{3} +
18\, (-1)^{n}\, \mathcal{A}_{n}\,\mathcal{B}_{n} -27.\label{ram-2-w3}
\end{align}

We note that
\begin{equation}\label{ram-2-w4}
\mathcal{B}_{n} =
\big( 2\, \cos \alpha \big)^{n} +
\big( 2\, \cos 2\alpha \big)^{n} +
\big( 2\, \cos 4\alpha \big)^{n},
\end{equation}
so we get the following interesting identity
\begin{equation}\label{ram-2-w5}
\frac{1}{3}\, \Big(6 + \mathcal{B}_{3n} - \mathcal{B}_{n}^{3}\Big) =
\sqrt[3]{ \tfrac{1}{2} \big(\mathcal{S}_{3n,1} + \sqrt{\mathcal{T}_{3n,1}}\,\big) }
+
\sqrt[3]{ \tfrac{1}{2} \big(\mathcal{S}_{3n,1} - \sqrt{\mathcal{T}_{3n,1}}\,\big) }.
\end{equation}
Moreover, if $n\in \mathbb{N}$ then by~\cite[eq.~(3.34)]{WitulaSlota-A7s}
and by~(\ref{nr-alpha}) and~(\ref{nr-delta}) we obtain
\begin{multline}\label{ram-2-w1-nw}
\big( 2\, \cos \alpha \big)^{-n/3} +
\big( 2\, \cos 2\alpha \big)^{-n/3} +
\big( 2\, \cos 4\alpha \big)^{-n/3} = {}\\
{}=
\sqrt[3]{
(-1)^{n}\, \mathcal{A}_{n} + 6
- \frac{3}{\sqrt[3]{2}} \,
\Big(
\sqrt[3]{ \mathcal{S}_{n,1} + \sqrt{\mathcal{T}_{n,1}} }
+
\sqrt[3]{ \mathcal{S}_{n,1} - \sqrt{\mathcal{T}_{n,1}} }\,\,
\Big)
}.
\end{multline}
Hence, for example for $n=2$ the formula~(\ref{sf-w1}) follows.

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\textbf{2.}
By~\cite[eq.~(3.27)]{WitulaSlota-A7s} we obtain
\begin{multline}\label{w2-d}
\sqrt[3]{ 2\, \sin \alpha\,
\big(2\, \cos 4\alpha \big)^{n} } +
\sqrt[3]{ 2\, \sin 2\alpha\,
\big(2\, \cos \alpha \big)^{n} } +
\sqrt[3]{ 2\, \sin 4\alpha\,
\big(2\, \cos 2\alpha \big)^{n} }
= {}\\
{}=
\sqrt[3]{
p_{n,2} - 6 \, \sqrt[6]{7}
- \frac{3}{\sqrt[3]{2}} \,
\Big(
\sqrt[3]{ \mathcal{S}_{n,2} + \sqrt{\mathcal{T}_{n,2}} }
+
\sqrt[3]{ \mathcal{S}_{n,2} - \sqrt{\mathcal{T}_{n,2}} }\,\,
\Big)
},
\end{multline}
where
\begin{align}
\mathcal{S}_{n,2} &=
7\,(-1)^{n}\, {B}_{n}\, \big( 6\, \sqrt[6]{7} - p_{n,2} \big)
-6\, \sqrt[3]{7} \, p_{n,2} +9 \, \sqrt{7},\label{w2-e}\\
\mathcal{T}_{n,2} &=
49\, B_{n}^{2} \, \big( p_{n,2}^{2} - 28\, (-1)^n\, B_{n} \big) +{} \nonumber\\
&\phantom{==}
+2\, \sqrt{7}\, p_{n,2}\, \big( 2\, p_{n,2}^{2} - 63\, (-1)^n B_{n} \big) -189,
\label{w2-f}\\
%\end{align}
%\begin{align}
p_{n,2}&=
\left\{
\begin{array}{ll}
a_{n/2} & \mbox{ if $n$ is even},\\
\overline{\alpha}_{(n-1)/2} & \mbox{ if $n$ is odd},
\end{array}
\right.\label{w2-a}\\
q_{n,2}&= 7\, (-1)^n\, {B}_{n},\label{w2-b}\\
r_{n,2} &\equiv \sqrt{7}.\label{w2-c}
\end{align}


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\textbf{3.}
By~\cite[eq.~(3.28)]{WitulaSlota-A7s} we obtain
\begin{multline}\label{w3-d}
\sqrt[3]{ 2\, \sin \alpha\,
\big(2\, \cos 2\alpha \big)^{n} } +
\sqrt[3]{ 2\, \sin 2\alpha\,
\big(2\, \cos 4\alpha \big)^{n} } +
\sqrt[3]{ 2\, \sin 4\alpha\,
\big(2\, \cos \alpha \big)^{n} }
= {}\\
{}=
\sqrt[3]{
p_{n,3} - 6 \, \sqrt[6]{7}
- \frac{3}{\sqrt[3]{2}} \,
\Big(
\sqrt[3]{ \mathcal{S}_{n,3} + \sqrt{\mathcal{T}_{n,3}} }
+
\sqrt[3]{ \mathcal{S}_{n,3} - \sqrt{\mathcal{T}_{n,3}} }\,\,
\Big)
},
\end{multline}
where
\begin{align}
\mathcal{S}_{n,3} &=
7\,(-1)^{n}\, \big( {B}_{n} -{C}_{n} \big)\, \big( p_{n,3} - 6\, \sqrt[6]{7} \big)
-6\, \sqrt[3]{7} \, p_{n,3} +9\, \sqrt{7},\label{w3-e}\\
\mathcal{T}_{n,3} &=
49\, \big( {B}_{n} -{C}_{n} \big)^{2} \,
\big( p_{n,3}^{2} + 28\, (-1)^n\, \big( {B}_{n} -{C}_{n} \big) \big) +{} \nonumber\\
&\phantom{==}
+2\, \sqrt{7}\, p_{n,3}\, \big( 2\, p_{n,3}^{2} +
63\, (-1)^n \big( {B}_{n} -{C}_{n} \big) \big) - 189,
\label{w3-f}\\
%\end{align}
%\begin{align}
p_{n,3}&=
\left\{
\begin{array}{ll}
b_{n/2} & \mbox{ if $n$ is even},\\
\overline{\beta}_{(n-1)/2} & \mbox{ if $n$ is odd},
\end{array}
\right.\label{w3-a}\\
q_{n,3}&= 7\, (-1)^{n-1}\, \big( {B}_{n} -{C}_{n} \big),\label{w3-b}\\
r_{n,3} &\equiv \sqrt{7}.\label{w3-c}
\end{align}


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\textbf{4.}
By~\cite[eq.~(3.29)]{WitulaSlota-A7s} we obtain
\begin{multline}\label{w4-d}
\sqrt[3]{ 2\, \sin \alpha\,
\big(2\, \cos \alpha \big)^{n} } +
\sqrt[3]{ 2\, \sin 2\alpha\,
\big(2\, \cos 2\alpha \big)^{n} } +
\sqrt[3]{ 2\, \sin 4\alpha\,
\big(2\, \cos 4\alpha \big)^{n} }
= {}\\
{}=
\sqrt[3]{
p_{n,4} - 6 \, \sqrt[6]{7}
- \frac{3}{\sqrt[3]{2}} \,
\Big(
\sqrt[3]{ \mathcal{S}_{n,4} + \sqrt{\mathcal{T}_{n,4}} }
+
\sqrt[3]{ \mathcal{S}_{n,4} - \sqrt{\mathcal{T}_{n,4}} }\,\,
\Big)
},
\end{multline}
where
\begin{align}
\mathcal{S}_{n,4} &=
7\,(-1)^{n}\, {C}_{n}\, \big( p_{n,4} - 6\, \sqrt[6]{7} \big)
- 6\, \sqrt[3]{7} p_{n,4} + 9\, \sqrt{7} ,\label{w4-e}\\
\mathcal{T}_{n,4} &=
\big( 7\, p_{n,4}\, {C}_{n} + 9\, (-1)^{n}\, \sqrt{7} \big)^{2} \,
+4\, \big( 7^3\, (-1)^{n}\, C_{n}^{3} + \sqrt{7}\, p_{n,4}^{3} \big) -756,
\label{w4-f}\\
%\end{align}
%\begin{align}
p_{n,4}&=
\left\{
\begin{array}{ll}
c_{n/2} & \mbox{ if $n$ is even},\\
\overline{\gamma}_{(n-1)/2} & \mbox{ if $n$ is odd},
\end{array}
\right.\label{w4-a}\\
q_{n,4}&= 7\, (-1)^{n-1}\, {C}_{n},\label{w4-b}\\
r_{n,4} &\equiv \sqrt{7}.\label{w4-c}
\end{align}

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\textbf{5.}
By~\cite[eq.~(3.31)]{WitulaSlota-A7s} we obtain
\begin{multline}\label{w5-d}
\sqrt[3]{ 2\, \cos \alpha\,
\big( 2\, \cos 4\alpha \big)^{n} } +
\sqrt[3]{ 2\, \cos 2\alpha\,
\big( 2\, \cos \alpha \big)^{n} } +
\sqrt[3]{ 2\, \cos 4\alpha\,
\big( 2\, \cos 2\alpha \big)^{n} }
= {}\\
{}=
\sqrt[3]{
g_{n} + 6
- \frac{3}{\sqrt[3]{2}} \,
\Big(
\sqrt[3]{ \mathcal{S}_{n,5} + \sqrt{\mathcal{T}_{n,5}} }
+
\sqrt[3]{ \mathcal{S}_{n,5} - \sqrt{\mathcal{T}_{n,5}} }\,\,
\Big)
},
\end{multline}
where
\begin{align}
\mathcal{S}_{n,5} &=
-g_{n}\,q_{n,5} - 6  \, \big( g_{n} + q_{n,5} \big)-9,\label{w5-e}\\
\mathcal{T}_{n,5} &=
\big( g_{n}\, q_{n,5} +9 \big)^{2} -4\, \big( g_{n}^{3} + q_{n,5}^{3} \big) -108,
\label{w5-f}
\end{align}
\begin{align}
p_{n,5}&=-g_{n},\label{w5-a}\\
q_{n,5}&= (-1)^{n}\, \big( \mathcal{A}_{n} + \mathcal{A}_{n+1} - 7\, {A}_{n}\big),\label{w5-b}\\
r_{n,5} &\equiv -1.\label{w5-c}
\end{align}

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\textbf{6.}
By~\cite[eq.~(3.32)]{WitulaSlota-A7s} and~(\ref{nr-beta}) we obtain
\begin{multline}\label{w11-d}
\sqrt[3]{ \cos \alpha\,
\big( \sec 2\alpha \big)^{n} } +
\sqrt[3]{ \cos 2\alpha\,
\big( \sec 4\alpha \big)^{n} } +
\sqrt[3]{ \cos 4\alpha\,
\big( \sec \alpha \big)^{n} }
= {}\\
{}=
\sqrt[3]{
2^{n-1}\, \Big( {-}p_{n,6} + 6
- \frac{3}{\sqrt[3]{2}} \,
\Big(
\sqrt[3]{ \mathcal{S}_{n,6} + \sqrt{\mathcal{T}_{n,6}} }
+
\sqrt[3]{ \mathcal{S}_{n,6} - \sqrt{\mathcal{T}_{n,6}} }\,\,
\Big) \Big)
},
\end{multline}
where
\begin{align}
\mathcal{S}_{n,6} &=
(-1)^n\, \big( \mathcal{A}_{n+1} - \mathcal{A}_{n}\big)\,
\big( \mathcal{B}_{n+2} - \mathcal{B}_{n} - 6 \big)
+ 6\,\big( \mathcal{B}_{n+2} - \mathcal{B}_{n}\big)-9,\label{w11-e}\\
\mathcal{T}_{n,6} &=
\big( \mathcal{B}_{n+2} - \mathcal{B}_{n}\big)^2\,
\Big( \big( \mathcal{A}_{n} - \mathcal{A}_{n+1}\big)^2 +
4\, \big( \mathcal{B}_{n+2} - \mathcal{B}_{n}\big) \Big) +{}\nonumber\\
&\phantom{=}+
2\, (-1)^{n}\,\big( \mathcal{A}_{n} - \mathcal{A}_{n+1}\big)\,
\Big( 2\, \big( \mathcal{A}_{n} - \mathcal{A}_{n+1}\big)^2 + %{}\\
%&\phantom{=}+
9\, \big( \mathcal{B}_{n+2} - \mathcal{B}_{n}\big) \Big)
- 27,\label{w11-f}\\
%\end{align}
%\begin{align}
p_{n,6}&=(-1)^n\, \big( \mathcal{A}_{n} - \mathcal{A}_{n+1}\big),\label{w11-a}\\
q_{n,6}&= f_n + h_n = \mathcal{B}_{n} - \mathcal{B}_{n+2},\label{w11-b}\\
r_{n,6} &\equiv -1.\label{w11-c}
\end{align}


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\textbf{7.}
By~\cite[eq.~(3.33)]{WitulaSlota-A7s} and~(\ref{nr-gamma}) we obtain
\begin{multline}\label{w10-d}
\sqrt[3]{ \cos \alpha\,
\big( \sec 4\alpha \big)^{n} } +
\sqrt[3]{ \cos 2\alpha\,
\big( \sec \alpha \big)^{n} } +
\sqrt[3]{ \cos 4\alpha\,
\big( \sec 2\alpha \big)^{n} }
= {}\\
{}=
\sqrt[3]{
2^{n-1}\, \Big( {-}p_{n,7} + 6
- \frac{3}{\sqrt[3]{2}} \,
\Big(
\sqrt[3]{ \mathcal{S}_{n,7} + \sqrt{\mathcal{T}_{n,7}} }
+
\sqrt[3]{ \mathcal{S}_{n,7} - \sqrt{\mathcal{T}_{n,7}} }\,\,
\Big) \Big)
},
\end{multline}
where
\begin{align}
\mathcal{S}_{n,7} &=
(-1)^n\, \big( \mathcal{A}_{n+1} - \mathcal{A}_{n-1}\big)\,
\big( 6 - \mathcal{B}_{n+1} - \mathcal{B}_{n} \big)
+ 6\,\big( \mathcal{B}_{n+1} + \mathcal{B}_{n}\big)-9,\label{w10-e}\\
\mathcal{T}_{n,7} &=
\big( \mathcal{B}_{n+1} + \mathcal{B}_{n}\big)^2\,
\Big( \big( \mathcal{A}_{n+1} - \mathcal{A}_{n-1}\big)^2 +
4\, \big( \mathcal{B}_{n+1} + \mathcal{B}_{n}\big) \Big) +{}\nonumber\\
&+
2\, (-1)^{n}\,\big( \mathcal{A}_{n+1} - \mathcal{A}_{n-1}\big)\,
\Big( 2\, \big( \mathcal{A}_{n+1} - \mathcal{A}_{n-1}\big)^2 +
9\, \big( \mathcal{B}_{n+1} + \mathcal{B}_{n}\big) \Big)
- 27,\label{w10-f}\\
%\end{align}
%\begin{align}
p_{n,7}&=(-1)^n\, \big( \mathcal{A}_{n+1} - \mathcal{A}_{n-1}\big),\label{w10-a}\\
q_{n,7}&= f_n + g_n = - \mathcal{B}_{n} - \mathcal{B}_{n+1},\label{w10-b}\\
r_{n,7} &\equiv -1.\label{w10-c}
\end{align}


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\textbf{8.}
By~\cite[eq.~(3.30)]{WitulaSlota-A7s}, we obtain
\begin{multline}\label{w6-d}
\sqrt[3]{ 2\, \cos \alpha\,
\big( 2\, \cos 2\alpha \big)^{n} } +
\sqrt[3]{ 2\, \cos 2\alpha\,
\big( 2\, \cos 4\alpha \big)^{n} } +
\sqrt[3]{ 2\, \cos 4\alpha\,
\big( 2\, \cos \alpha \big)^{n} }
= {}\\
{}=
\sqrt[3]{
f_{n} + 6
- \frac{3}{\sqrt[3]{2}} \,
\Big(
\sqrt[3]{ \mathcal{S}_{n,8} + \sqrt{\mathcal{T}_{n,8}} }
+
\sqrt[3]{ \mathcal{S}_{n,8} - \sqrt{\mathcal{T}_{n,8}} }\,\,
\Big)
},
\end{multline}
where
\begin{align}
\mathcal{S}_{n,8} &=
-f_{n}\,q_{n,8} - 6  \, \big( f_{n} + q_{n,8} \big)-9,\label{w6-e}\\
\mathcal{T}_{n,8} &=
\big( f_{n}\, q_{n,8} +9 \big)^{2} -4\, \big( f_{n}^{3} + q_{n,8}^{3} \big) -108,
\label{w6-f}\\
%\end{align}
%\begin{align}
p_{n,8}&=-f_{n},\label{w6-a}\\
q_{n,8}&= (-1)^{n}\, \big( 7\, {A}_{n} - 3\, \mathcal{A}_{n} \big)
=(-1)^n \big( \mathcal{A}_{n} - \mathcal{A}_{n-2} \big),\label{w6-b}\\
r_{n,8} &\equiv -1.\label{w6-c}
\end{align}

\begin{remark}\label{rem4.3}
As results from direct observation of the value of expression
$$
\sqrt[3]{ \frac{1}{2}\, \Big(\mathcal{S}_{n,8} - \sqrt{\mathcal{T}_{n,8}}\Big) }
$$
for $n=0,1,\ldots,2000$, for the indicated index values, the following
equalities hold
$$
\sqrt[3]{ \frac{1}{2}\, \Big(\mathcal{S}_{n,8} - \sqrt{\mathcal{T}_{n,8}}\Big) } =
\left\{
\begin{array}{ll}
(-1)^{k-1}\, \widehat{x}_{k}, &\quad \mbox{ for }\  n=3\, k-1 \geq 5,\\[0.8ex]
\sqrt[3]{7}\, (-1)^{k}\, \widehat{y}_{k}, &\quad \mbox{ for }\ n=3\, k,\\[0.8ex]
\sqrt[3]{49}\, (-1)^{k+1}\, \widehat{z}_{k}, &\quad \mbox{ for }\ n=3\, k+1,\\
\end{array}
\right.
$$
for $k=1,2,\ldots$, where
\begin{align*}
\widehat{x}_{1} &= 2, & \widehat{x}_{2} &= 5, & \widehat{x}_{3} &=16,\\
\widehat{y}_{1} &= 1, & \widehat{y}_{2} &= 4, & \widehat{y}_{3} &=12,\\
\widehat{z}_{1} &= 1, & \widehat{z}_{2} &= 3, & \widehat{z}_{3} &=9,
\end{align*}
and the elements of any of the sequences: $\{\widehat{x}_{k}\}_{k=1}^{\infty}$,
$\{\widehat{y}_{k}\}_{k=1}^{\infty}$ and $\{\widehat{z}_{k}\}_{k=1}^{\infty}$,
satisfy the following recurrent relation
$$
\mathbb{X}_{n+3} - 4\, \mathbb{X}_{n+2} + 3\, \mathbb{X}_{n+1} +\mathbb{X}_{n} =0.
$$
Also, the following interesting relationships occur (see~(\ref{aa})):
\begin{align*}
\widehat{x}_{k} &= A_{k}(-1) + 2\, C_{k}(-1) - C_{k-2}(-1),\qquad k\geq 2,\\
\widehat{y}_{k} &= A_{k}(-1) + C_{k}(-1),\\
\widehat{z}_{k} &= C_{k}(-1) - B_{k}(-1).
\end{align*}
Hence, by~\cite[eqs.~(3.17), (3.18), (3.19)]{WitulaSlota-A7}, we obtain,
inter alia, the following Binet formulas:
\begin{multline*}
\widehat{x}_{k} =
 \big( 2 - 2\, \cos \alpha +
4\, \cos 2\alpha \big)\,
\big( 1 - 2\, \cos \alpha \big)^{k-2} +{}\\
{}+
 \big( 2 - 2\, \cos 2\alpha +
4\, \cos 4\alpha \big)\,
\big( 1 - 2\, \cos 2\alpha \big)^{k-2} +{}\\
{}+
 \big( 2 - 2\, \cos 4\alpha +
4\, \cos \alpha \big)\,
\big( 1 - 2\, \cos 4\alpha \big)^{k-2},
\end{multline*}
\begin{multline*}
\widehat{y}_{k} =
\frac{2}{7}\, \big( 1 + \cos 2\alpha -
2\, \cos 4\alpha \big)\,
\big( 1 - 2\, \cos \alpha \big)^{k} +{}\\
{}+
\frac{2}{7}\, \big( 1 + \cos 4\alpha -
2\, \cos \alpha \big)\,
\big( 1 - 2\, \cos 2\alpha \big)^{k} +{}\\
{}+
\frac{2}{7}\, \big( 1 + \cos \alpha -
2\, \cos 2\alpha \big)\,
\big( 1 - 2\, \cos 4\alpha \big)^{k} ,
\end{multline*}
\begin{multline*}
\widehat{z}_{k} =
\frac{2}{7}\,
\big( \cos 2\alpha -  \cos \alpha \big)\,
\big( 1 - 2\, \cos \alpha \big)^{k} + {}\\
{}+
\frac{2}{7}\,
\big( \cos 4\alpha -  \cos 2\alpha \big)\,
\big( 1 - 2\, \cos 2\alpha \big)^{k} +{}\\
{}+
\frac{2}{7}\,
\big( \cos \alpha -  \cos 4\alpha \big)\,
\big( 1 - 2\, \cos 4\alpha \big)^{k}.
\end{multline*}

Next, as results from direct observation of the value of expression
$$
\sqrt[3]{ \frac{1}{2}\, \Big(\mathcal{S}_{n,8} +
\sqrt{\mathcal{T}_{n,8}}\Big) }
$$
for $n=0,1,\ldots,2000$, for the indicated index values, the following equations hold
$$
\sqrt[3]{ \frac{1}{2}\, \Big(\mathcal{S}_{n,8} +
\sqrt{\mathcal{T}_{n,8}}\Big) } = \left\{
\begin{array}{ll}
\widetilde{x}_{k}, &\quad \mbox{ for }\  n=3\, k+2,\\[0.8ex]
\sqrt[3]{7}\, \widetilde{y}_{k}, &\quad \mbox{ for }\ n=3\, k+1,\\[0.8ex]
\sqrt[3]{49}\, \widetilde{z}_{k}, &\quad \mbox{ for }\ n=3\, k,\\
\end{array}
\right.
$$
for $k=1,2,\ldots$, where $x_0=2$ and
\begin{align*}
\widetilde{x}_{1} &= 8, & \widetilde{x}_{2} &= 29, & \widetilde{x}_{3} &=120,\\
\widetilde{y}_{1} &= 1, & \widetilde{y}_{2} &= 2, & \widetilde{y}_{3} &=10,\\
\widetilde{z}_{1} &= 1, & \widetilde{z}_{2} &= 3, & \widetilde{z}_{3} &=13,
\end{align*}
and the elements of any of the sequences
$\{\widetilde{x}_{k}\}_{k=2}^{\infty}$,
$\{\widetilde{y}_{k}\}_{k=1}^{\infty}$ and
$\{\widetilde{z}_{k}\}_{k=1}^{\infty}$
satisfy the following recurrent relation:
$$
\mathbb{X}_{n+3} - 3\, \mathbb{X}_{n+2} - 4\, \mathbb{X}_{n+1}
-\mathbb{X}_{n} =0.
$$
After substitution $x\mapsto (x-1)$ in the respective characteristic polynomial
of this relation, we obtain polynomial $x^3-7\, x-7$, for which
by~\cite[eq.~(4.14)]{WitulaSlota-A7s}, for $n=1$, we obtain
\begin{equation*}
\mathbb{X}^3 -7 \, \mathbb{X} - 7 =
\prod_{k=0}^{2} \Big( \mathbb{X} + \frac{\sqrt{7}}{2}\,
\csc (2^k\alpha) \Big).
\end{equation*}
Hence, the following Binet formulas hold
\begin{equation*}
{p}_{n} =
a_p\, \big( 1 - \frac{\sqrt{7}}{2}\, \csc \alpha\big)^{\! n} +
b_p\, \big( 1 - \frac{\sqrt{7}}{2}\, \csc 2\alpha\big)^{\! n} +
c_p\, \big( 1 - \frac{\sqrt{7}}{2}\, \csc 4\alpha\big)^{\! n},
\end{equation*}
for every $p\in \{\widetilde{x},\widetilde{y},\widetilde{z}\}$, and where
\begin{align*}
a_{\tilde{x}} &\approx -1.246979604,  & b_{\tilde{x}} &\approx  0.4450418679, & c_{\tilde{x}} &\approx 1.801937736,\\
a_{\tilde{y}} &\approx -1.064961507,  & b_{\tilde{y}} &\approx 0.9189943261,  & c_{\tilde{y}} &\approx 0.1459671806,\\
a_{\tilde{z}} &\approx -0.4355596199, & b_{\tilde{z}} &\approx 0.2417173531,  & c_{\tilde{z}} &\approx 0.1938422668.
\end{align*}
Additionally, we note that
\begin{align*}
\widetilde{x}_{0} &= a_{\tilde{x}} + b_{\tilde{x}} + c_{\tilde{x}} = 1,\\
\widetilde{y}_{0} &= a_{\tilde{y}} + b_{\tilde{y}} + c_{\tilde{y}} = 0,\\
\widetilde{z}_{0} &= a_{\tilde{z}} + b_{\tilde{z}} + c_{\tilde{z}} = 0.
\end{align*}
By juxtaposing the obtained values of expressions
$$
\sqrt[3]{ \frac{1}{2}\, \Big(\mathcal{S}_{n,8} \pm
\sqrt{\mathcal{T}_{n,8}}\Big) }
$$
we can now invest formula~(\ref{w6-d}) to the new interesting form
(for cases $n=3k, 3k+1, 3k+2$ respectively, and only within the
indicated range of values $n=1,2,\ldots,2000$):
\begin{multline*}
\sqrt[3]{2\, \cos \alpha}\,
\big( 2\, \cos 2\alpha\big)^{k} +
\sqrt[3]{2\, \cos 2\alpha}\,
\big( 2\, \cos 4\alpha\big)^{k} +
\sqrt[3]{2\, \cos 4\alpha}\,
\big( 2\, \cos \alpha\big)^{k} ={}\\
{}=
\sqrt[3]{ f_{3k} + 6 + 3\, (-1)^{k-1} \widehat{y}_{k}\, \sqrt[3]{7} -
3\, \widetilde{z}_{k}\, \sqrt[3]{49}}\,,
\end{multline*}
\begin{multline*}
\sqrt[3]{\sec 4\alpha}\,
\big( 2\, \cos 2\alpha \big)^{k} +
\sqrt[3]{\sec \alpha}\,
\big( 2\, \cos 4\alpha \big)^{k} +
\sqrt[3]{\sec 2\alpha}\,
\big( 2\, \cos \alpha \big)^{k} ={}\\
{}=
\sqrt[3]{ 2\, \big(f_{3k+1} + 6 - 3\,  \widetilde{y}_{k}\, \sqrt[3]{7} +
3\,(-1)^{k}\, \widehat{z}_{k}\, \sqrt[3]{49}\big)}\,,
\end{multline*}
and the formula which is equivalent to relation~(\ref{w-nr2})
and which generates the identity
\begin{equation}
7\, \psi_{k}^{3} =
f_{3k+2} + 6 - 3\, \big( \widetilde{x}_{k} + (-1)^{k}\, \widehat{x}_{k+1} \big).
\end{equation}
\end{remark}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\textbf{9.}
By~\cite[eq.~(4.15)]{WitulaSlota-A7s} we obtain
\begin{multline}\label{w7-d}
\big( 2\, \sin \alpha \big)^{n/3}  +
\big( 2\, \sin 2\alpha \big)^{n/3}  +
\big( 2\, \sin 4\alpha \big)^{n/3}
= {}\\
{}=
\sqrt[3]{
z_{n-1} + 6\, (-1)^{n}\, 7^{n/6}
- \frac{3}{\sqrt[3]{2}} \,
\Big(
\sqrt[3]{ \mathcal{S}_{n,9} + \sqrt{\mathcal{T}_{n,9}} }
+
\sqrt[3]{ \mathcal{S}_{n,9} - \sqrt{\mathcal{T}_{n,9}} }\,\,
\Big)
},
\end{multline}
where
\begin{align}
\mathcal{S}_{n,9} &=
\big( \tfrac{1}{2}\, z_{n-1} + 3\, (-1)^{n}\, 7^{n/6} \big)\,
\big( z_{2n-1} - z_{n-1}^{2} \big) - \nonumber\\
&\phantom{==}
- 6\cdot 7^{n/3} \, z_{n-1} - 9\, (-1)^{n}\, 7^{n/2},\label{w7-e}\\
\mathcal{T}_{n,9} &=
(-1)^{n}\, 7^{n/2}\, z_{n-1}\, \big( 5\, z_{n-1}^{2} - 9\, z_{2n-1} \big) +\nonumber\\
&\phantom{==}
+ \tfrac{1}{4}\, \big( 2\, z_{2n-1} - z_{n-1}^{2} \big)\,
\big( z_{n-1}^{2} - z_{2n-1} \big)^{2} - 27 \cdot 7^{n},
\label{w7-f}\\
%\end{align}
%$$
p_{n,9}&=-z_{n-1},\quad
q_{n,9}= \tfrac{1}{2} \, \big( {z}_{n-1}^{2} - z_{2n-1} \big),\quad
r_{n,9}= (-1)^{n-1}\, 7^{n/2}.
%$$
\end{align}
%\begin{align}
%p_{n,7}&=-z_{n-1},\label{w7-a}\\
%q_{n,7}&= \tfrac{1}{2} \, \big( {z}_{n-1}^{2} - z_{2n-1} \big),\label{w7-b}\\
%r_{n,7} &= (-1)^{n-1}\, 7^{n/2}.\label{w7-c}
%\end{align}


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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\textbf{10.}
By~\cite[eq.~(2.2)]{WitulaSlota-A7s} we have (for $\delta\in \mathbb{R}$):
\begin{multline}\label{ram-2-w6}
\big( 1 + 2\, \delta\, \cos \alpha \big)^{n/3} +
\big( 1 + 2\, \delta\, \cos 2\alpha \big)^{n/3} +
\big( 1 + 2\, \delta\, \cos 4\alpha \big)^{n/3} = {}\\
{}=
\bigg[
\mathcal{A}_{n}(\delta) + 6\, \widehat{\delta}^{\,n/3} -
 \frac{3}{\sqrt[3]{2}} \,
\Big(
\sqrt[3]{ \mathcal{S}_{n}(\delta) + \sqrt{\mathcal{T}_{n}(\delta)} }
+
\sqrt[3]{ \mathcal{S}_{n}(\delta) - \sqrt{\mathcal{T}_{n}(\delta)} }\,\,
\Big)
\bigg]^{1/3},
\end{multline}
where
\begin{align}
\widehat{\delta} &= \delta^3-2\, \delta^2-\delta+1,\label{ram-2-w7}\\
\mathcal{S}_{n}(\delta) &=
-\mathcal{A}_{n}(\delta)\, \mathcal{B}_{n}(\delta)
- 6\, \mathcal{B}_{n}(\delta)\, \widehat{\delta}^{\,n/3}
- 6\, \mathcal{A}_{n}(\delta)\, \widehat{\delta}^{\,2n/3}
- 9\, \widehat{\delta}^{\,n},\label{ram-2-w8}\\
\mathcal{T}_{n}(\delta) &=
\mathcal{A}_{n}^{2}(\delta)\,\mathcal{B}_{n}^{2}(\delta)
- 4\, \mathcal{A}_{n}^{3}(\delta)\, \widehat{\delta}^{\,n}
- 4\,\mathcal{B}_{n}^{3}(\delta)%\nonumber\\
%&\phantom{==}
+ 18\, \mathcal{A}_{n}(\delta)\,\mathcal{B}_{n}(\delta)\, \widehat{\delta}^{\,n}
- 27\, \widehat{\delta}^{\,2\, n}\label{ram-2-w9aa}\\
&=
\big(\mathcal{A}_{n}(\delta)\,\mathcal{B}_{n}(\delta) + 9\, \widehat{\delta}^{\,n} \big)^2
-4\, \big( \mathcal{A}_{n}^{3}(\delta)\, \widehat{\delta}^{\,n}
+\mathcal{B}_{n}^{3}(\delta) \big) - 108\, \widehat{\delta}^{\,2\, n}.\label{ram-2-w9}
\end{align}



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\textbf{11.}
By~\cite[eq.~(6.14)]{WitulaSlota-A7s} we have
\begin{multline}\label{ram-2-w10}
7^{n/6}\,
\big(
(  \cot \alpha )^{n/3}\,+
(  \cot 2\alpha )^{n/3}\,+
(  \cot 4\alpha )^{n/3} \big)
 = {}\\
{}=
\bigg[ 3^n\, \mathcal{A}_{n}(\tfrac{2}{3}) + 6\, (-1)^{n}\, 7^{n/3} +
 \frac{3}{\sqrt[3]{2}} \,  7^{n/3}\,
\Big( \sqrt[3]{ \mathcal{S}_{n}^{'} + \sqrt{\mathcal{T}_{n}^{'}} } +
\sqrt[3]{ \mathcal{S}_{n}^{'} - \sqrt{\mathcal{T}_{n}^{'}} }\,\, \Big) \bigg]^{1/3},
\end{multline}
where
\begin{align}
\mathcal{S}_{n}^{'} &=
\big( 3^n\, \mathcal{A}_{n}(\tfrac{2}{3}) + 6\,(-1)^{n}\,  7^{n/3} \big)\,
\Omega_{n} \big( \tfrac{2\, i}{\sqrt{7}}\big) %+ {}\nonumber\\
%&\phantom{==}
+ 6\, \big( \tfrac{3}{\sqrt[3]{7}} \big)^{n}\,  \mathcal{A}_{n}(\tfrac{2}{3})
+ 9\, (-1)^{n},
\label{ram-2-w11}\\
\mathcal{T}_{n}^{'} &=
%3^{2n}\, \mathcal{A}_{n}^{2}(\tfrac{2}{3})\, \Omega_{n}^{2} \big( \tfrac{2\, i}{\sqrt{7}}\big)
%+18\, (-3)^{n}\, \mathcal{A}_{n}(\tfrac{2}{3})\, \Omega_{n} \big( \tfrac{2\, i}{\sqrt{7}}\big) -{}\nonumber\\
%&\phantom{==}
%- 4\, \big( 7^n\, \Omega_{n}^{3} \big( \tfrac{2\, i}{\sqrt{7}}\big)
%+ (-27)^{n}\, 7^{-n}\, \mathcal{A}_{n}^{3}(\tfrac{2}{3}) \big) -27 \nonumber\\
%&=
\big(
(-3)^{n}\, \mathcal{A}_{n}(\tfrac{2}{3})\, \Omega_{n} \big( \tfrac{2\, i}{\sqrt{7}}\big)
+9
\big)^{2} %- {}\nonumber\\
%&\phantom{==}
- 4\, \big( 7^n\, \Omega_{n}^{3} \big( \tfrac{2\, i}{\sqrt{7}}\big)
+ \big(-\tfrac{27}{7} \big)^{n}\, \mathcal{A}_{n}^{3}(\tfrac{2}{3}) \big) - 108.
\label{ram-2-w12}
\end{align}
The numbers $\Omega_{n}(\delta)$ are defined
for $n\in \mathbb{N}$ and $\delta\in \mathbb{C}$,
in the following way
$$
\Omega_{n}(\delta) :=
\big( 1+2\, i\, \delta\, \sin\alpha \big)^{n} +
\big( 1+2\, i\, \delta\, \sin 2\alpha \big)^{n} +
\big( 1+2\, i\, \delta\, \sin 4\alpha \big)^{n},
$$
(see~\cite[Section~6]{WitulaSlota-A7s} for more details).

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\begin{remark}
Moreover, we have
\begin{multline}\label{gwiazdka}
\big(\mathbb{X}- ( \tan \alpha )^{n}\big)
\big(\mathbb{X}- ( \tan 2\alpha )^{n}\big)
\big(\mathbb{X}- ( \tan 4\alpha )^{n}\big)
 = {}\\
{} = \mathbb{X}^3 - ({-}\sqrt{7} )^{n} \,
\Omega_{n} \big(\tfrac{2\, i}{\sqrt{7}}\big)\, \mathbb{X}^2 +
(-3)^n\,
\mathcal{A}_{n}(\tfrac{2}{3})\, \mathbb{X} - ({-}\sqrt{7})^{n}.
\end{multline}

This "distribution" easily results from~\cite[eq.~(6.14)]{WitulaSlota-A7s}.
Now we will present a~direct proof of the relation~(\ref{gwiazdka}),
because the formula~(6.14) in~\cite{WitulaSlota-A7s}
is presented without a~proof.
For this purpose, let us suppose that $\xi = \exp ( i\, 2\, \pi/7)$.
Then we have
\begin{multline*}
( \tan \alpha  )^{n}+
( \tan 2\alpha )^{n}+
( \tan 4\alpha )^{n}
={}\\
{}=
\Big( {-}i\, \frac{\xi - \xi^6}{\xi + \xi^6} \Big)^n +
\Big( {-}i\, \frac{\xi^2 - \xi^5}{\xi^2 + \xi^5} \Big)^n +
\Big( {-}i\, \frac{\xi^4 - \xi^3}{\xi^4 + \xi^3} \Big)^n
={}\\
{}=
\Big( \frac{-i}{(\xi + \xi^6)(\xi^2 + \xi^5)(\xi^4 + \xi^3)} \Big)^{n}\,
\bigg[
\Big( (\xi - \xi^6)(\xi^2 + \xi^5)(\xi^4 + \xi^3) \Big)^{n} +{}\\
{}+
\Big( (\xi^2 - \xi^5)(\xi + \xi^6)(\xi^4 + \xi^3) \Big)^{n} +
\Big( (\xi^4 - \xi^3)(\xi + \xi^6)(\xi^2 + \xi^5) \Big)^{n}
\bigg]={}%\\
\end{multline*}
\begin{multline*}
{}\stackrel{\mbox{\small \cite[eq.~(1.4)]{WitulaSlota-A7s}}}{=}
(-i)^n\,
\bigg[
\Big( 2\,(\xi^2 - \xi^5) - (\xi - \xi^6) -(\xi^2 - \xi^5) - (\xi^4 - \xi^3) \Big)^{n} +{}\\
{}+
\Big( 2\,(\xi^4 - \xi^3) - (\xi - \xi^6) -(\xi^2 - \xi^5) - (\xi^4 - \xi^3) \Big)^{n} +{}\\
{}+
\Big( 2\,(\xi - \xi^6) - (\xi - \xi^6) -(\xi^2 - \xi^5) - (\xi^4 - \xi^3) \Big)^{n}
\bigg] ={}\\
{}
\stackrel{\mbox{\small \cite[eq.~(1.1)]{WitulaSlota-A7s}}}{=}
\bigg[
\Big( {-}2\, i\, (\xi^2-\xi^5) - \sqrt{7} \Big)^{n} +
\Big( {-}2\, i\, (\xi^4-\xi^3) - \sqrt{7} \Big)^{n} +{}\\
{}+
\Big( {-}2\, i\, (\xi-\xi^6) - \sqrt{7} \Big)^{n}
\bigg]
=
\big( {-}\sqrt{7} \big)^{n}\,
\Omega_{n} \big(\tfrac{2\, i}{\sqrt{7}}\big),
\end{multline*}
and
\begin{multline*}
\big( \tan \alpha\,\tan 2\alpha \big)^{n}+
\big( \tan \alpha\,\tan 4\alpha \big)^{n}+
\big( \tan 2\alpha\,\tan 4\alpha \big)^{n}
={}\\
{}=
\Big(
\Big({-}i\, \frac{\xi - \xi^6}{\xi + \xi^6}\Big)\,
\Big({-}i\, \frac{\xi^2 - \xi^5}{\xi^2 + \xi^5}\Big)
\Big)^n +
\Big(
\Big({-}i\, \frac{\xi - \xi^6}{\xi + \xi^6}\Big)\,
\Big({-}i\, \frac{\xi^4 - \xi^3}{\xi^4 + \xi^3}\Big)
\Big)^n +{}\\
{}+
\Big(
\Big({-}i\, \frac{\xi^2 - \xi^5}{\xi^2 + \xi^5}\Big)\,
\Big({-}i\, \frac{\xi^4 - \xi^3}{\xi^4 + \xi^3}\Big)
\Big)^n
={}\\
%\end{multline*}
%\begin{multline*}
{}=
\Big((\xi + \xi^6)(\xi^2 + \xi^5)(\xi^4 + \xi^3)\Big)^{-n}\,
\bigg[
\Big( (\xi^6 - \xi)(\xi^2 - \xi^5)(\xi^4 + \xi^3) \Big)^{n} +{}\\
{}+
\Big( (\xi^6 - \xi)(\xi^4 - \xi^3)(\xi^2 + \xi^5) \Big)^{n} +
\Big( (\xi^5 - \xi^2)(\xi^4 - \xi^3)(\xi + \xi^6) \Big)^{n}
\bigg]
={}\\
{}=
\Big(\! {-}3 - 2\, (\xi+\xi^6)\! \Big)^{n} +
\Big(\! {-}3 - 2\, (\xi^4+\xi^3)\! \Big)^{n} +
\Big(\! {-}3 - 2\, (\xi^2+\xi^5)\! \Big)^{n}
=
(-3)^n\, \mathcal{A}_{n}(\tfrac{2}{3}).
\end{multline*}
\end{remark}


\noindent
\textbf{Final remark.}
I~was only after I~received the referee report on my paper that
I~learnt about two important publications in this field~\cite{Shevelev2007,Shevelev2009}.
Certainly, both papers supplement and enrich the contents of Section~\ref{roz3a}.
As a~spontaneous reaction to~\cite{Shevelev2007} and the report on the
present paper, two more papers sprang up~\cite{Witula-RCP}
and~\cite{Witula-Sup}.


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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Acknowledgments}


The author wish to express their gratitude to the Reviewer
for several helpful comments concerning the first version of my paper.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{table}[htb!]


\begin{center}
{\small%
\caption{}\label{tabela}
\medskip

\begin{tabular}{||c||r|r|r|r|r|r|r|r|r|r|r|r||}\hline\hline
$n$  &
$0$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ & $9$ & $10$ & $11$ \\ \hline\hline
$\psi_{n}$ &
$-1$ & $0$ & $-3$ & $2$ & $-8$ & $9$ & $-23$ & $33$ & $-70$ & $113$ & $-220$ & $376$\\ \hline
$\varphi_{n}$ &
$0$ & $-1$ & $1$ & $-3$ & $4$ & $-9$ & $14$ & $-28$ & $47$ & $-89$ & $155$ & $-286$\\\hline\hline
\end{tabular}
}
\end{center}

\end{table}



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%%%%%%%%%%%%%%               %%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%   Section     %%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%               %%%%%%%%%%%%%%%%%%
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\begin{thebibliography}{10}

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\textit{J.~Integer Seq.}  \textbf{10} (2007), 
\htmladdnormallink{Article 07.5.6}{http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Slota/witula13.html}.

\bibitem{WitulaSlota-A11}
R.~Witu{\l}a and D.~S{\l}ota,
{Quasi-Fibonacci numbers of order~11},
\textit{J.~Integer Seq.}  \textbf{10} (2007), 
\htmladdnormallink{Article 07.8.5}{http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Slota2/slota99.html}.

\bibitem{Witula-RCP}
R.~Witu{\l}a,
{Full description of Ramanujan cubic polynomials},
submitted.

\bibitem{Witula-Sup}
R.~Witu{\l}a,
{Ramanujan type trigonometric formulas
for arguments $\frac{2\, \pi}{7}$ and $\frac{2\, \pi}{9}$~--
supplement},
submitted.

\bibitem{Yaglom}
A.~M.~Yaglom and I.~M.~Yaglom,
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\end{thebibliography}



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\noindent 2000 {\it Mathematics Subject Classification}: Primary
11B37; Secondary 11B83, 11Y55, 33B10.

\noindent \emph{Keywords:} Ramanujan equalities, trigonometric recurrences.

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\noindent (Concerned with sequences \seqnum{A094648} and \seqnum{A006053}.)

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\vspace*{+.1in} \noindent Received July 6 2009; Revised version
received December 3 2009.
Published in {\it Journal of Integer Sequences}, December 3 2009.

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\noindent Return to \htmladdnormallink{Journal of Integer Sequences
home page}{http://www.math.uwaterloo.ca/JIS/}. \vskip .1in


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