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Abstract for R. L. Graham and B. D. Lubachevsky,
Repeated Patterns of Dense Packings of Equal Disks in a Square

\noindent
We examine sequences of dense packings
of $n$ congruent non-overlapping disks inside a square
which follow specific patterns as $n$ increases along
certain values, $n = n(1), n(2),... n(k),...$.
Extending and improving previous work
of Nurmela and  \"Osterg\aa rd  where
previous patterns for $n = n(k)$ of the form 
$ k^2$, $ k^2-1$, $k^2-3$, $k(k+1)$, and $4k^2+k$ were observed,
we identify new patterns for $n = k^2-2$
and $n = k^2+ \lfloor k/2 \rfloor$.
We also find denser packings than those in 
Nurmela and  \"Osterg\aa rd  
for $n =$21, 28, 34, 40, 43, 44, 45, and 47.
In addition, we produce what we conjecture to be
optimal packings for
$n =$51, 52, 54, 55, 56, 60, and 61.
Finally, for each identified sequence $n(1), n(2),... n(k),...$
which corresponds to some specific repeated pattern,
we identify a threshold index $k_0$, for which
the packing appears to be optimal for $k \le k_0$,
but for which the packing is not optimal (or does
not exist) for $k > k_0$.

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