Repeated Patterns of Dense Packings of Equal Disks in a Square

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 Östergård where previous patterns for $n = n(k)$ of the form k², k²-1, k²-3, $k(k+1)$, and 4k²+k were observed, we identify new patterns for n = k²-2 and n = k²+ k/2. We also find denser packings than those in Nurmela and Östergård 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 k₀, for which the packing appears to be optimal for k ≤ k₀, but for which the packing is not optimal (or does not exist) for k > k₀.