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 Ostergard where previous patterns for n = n(k) of the form k*k, k*k-1, k*k-3, k(k+1), and 4k*k+k were observed, we identify new patterns for n = k*k-2 and n = k*k+[k/2]. We also find denser packings than those in [Nurmela, Ostergard] 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 =< k_0, but for which the packing is not optimal (or does not exist) for k > k_0. 
Ronald L. Graham and Boris D. Lubachevsky