cp's OEIS Frontend

Sequence focuses on the nonsquare numbers in order to eliminate trivial solutions (A000217(n^2) = (n^2 + n^4)/2 = ((n + n^2)/2)^2 + ((n - n^2)/2)^2).

A073412 is a subsequence. Additionally, (A073412(n), A073412(n) + 1) gives consecutive pairs of this sequence that are (72, 73), (232, 233), (520, 521), (584, 585), ...

Proof:

Note that (a^2 + b^2)*(c^2 + d^2)/2 = ((a*c + b*d)^2 + (a*d - b*c)^2)/2 = ((a*c + b*d + a*d - b*c)/2)^2 + ((a*c + b*d - a*d + b*c)/2)^2 and A073412(n) + k is not a square by definition of it for 0 <= k <= 2. So this explains the reason of the fact that A073412(n) and A073412(n) + 1 are always members of this sequence. Furthermore, if there is a consecutive pair in this sequence, the lesser of pair must be in A073412 since n, n+1 and n+2 must be the sum of two nonzero squares, if n*(n+1)/2 and (n+1)*(n+2)/2 are the sum of two nonzero squares and n, n+1, n+2 are nonsquares. So exactly (A073412(n), A073412(n) + 1) gives consecutive pairs of this sequence.