A155985 a(1), then a(n) = smallest square not occurring earlier, not ending with zero and starting with the last digit of a(n-1).

1, 16, 64, 4, 49, 9, 961, 121, 144, 441, 169, 9025, 529, 9216, 625, 576, 676, 6084, 484, 4096, 6241, 196, 6561, 1024, 4225, 5041, 1089, 9409, 9604, 4356, 6724, 4489, 9801, 1156, 6889, 90601, 1225, 5184, 4624, 4761, 1296, 60025, 5329, 91204, 40401, 1369