A077222 a(1) = 1 and then alternately the smallest even and odd numbers not occurring earlier such that the sum of two successive terms is a square.

1, 8, 17, 32, 49, 72, 9, 16, 33, 48, 73, 96, 25, 24, 57, 64, 105, 120, 169, 56, 65, 104, 121, 168, 193, 248, 41, 40, 81, 88, 137, 152, 209, 80, 89, 136, 153, 208, 233, 128, 97, 192, 249, 112, 113, 176, 185, 256, 273, 352, 177, 184, 257, 272, 353, 376, 465, 160