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{\bf Abraham Isgur and Mustazee Rahman}
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{\bf On Variants of Conway and Conolly's Meta-Fibonacci Recursions}
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We study the recursions $A(n) = A(n-a-A^k(n-b)) + A(A^k(n-b))$ where
$a \geq 0$, $b \geq 1$ are integers and the superscript $k$ denotes a
$k$-fold composition, and also the recursion $C(n) = C(n-s-C(n-1)) +
C(n-s-2-C(n-3))$ where $s \geq 0$ is an interger. We prove that under
suitable initial conditions the sequences $A(n)$ and $C(n)$ will be
defined for all positive integers, and be monotonic with their forward
difference sequences consisting only of 0 and 1. We also show that the
sequence generated by the recursion for $A(n)$ with parameters
$(k,a,b) = (k,0,1)$, and initial conditions $A(1) = A(2) = 1$,
satisfies $A(E_n) = E_{n-1}$ where $E_n$ is a generalized Fibonacci
recursion defined by $E_n = E_{n-1} + E_{n-k}$ with $E_n = 1$ for $1
\leq n \leq k$.



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