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%I #26 Dec 11 2022 00:46:32
%S 1,25,49,121,169,289,729,784,841,961,1296,1681,2401,3969,5041,5184,
%T 5329,6561,6889,7744,8464,9801,10816,13689,18496,22201,32761,34969,
%U 40401,40804,41616,42436,44944,45796,46656,49729,51984,55696,66049,66564,72361,79524,85264
%N Perfect squares that are the sum of a perfect square and a factorial number.
%C For any factorial k = m*n where m > n and both m and n are even, ((m-n)/2)^2 + k = ((m+n)/2)^2 will appear in this sequence.
%e 13^2 = 7^2 + 5!.
%t With[{f = Range[9]!}, q[n_] := AnyTrue[f, IntegerQ @ Sqrt[n - #] &]; Select[Range[Floor @ Sqrt[f[[-1]]]]^2, q]] (* _Amiram Eldar_, Oct 28 2022 *)
%Y Cf. A000290, A000142.
%K nonn
%O 1,2
%A _Walter Robinson_, Oct 28 2022