Infinitely many $n$ such that $n, n+1, n+2$ are each the sum of two perfect squares. Prove that there exist infinitely many integers $n$ such that $n$, $n+1$, $n+2$ are all the sum of two perfect squares. Induction does not seem to be yielding any results.

One such infinite family is 
$$(2a^2 + 1)^2 - 1, (2a^2 + 1)^2 + 0^2, (2a^2 + 1)^2 + 1^2$$ for all integer $a$.
This works because 
$$(2a^2 + 1)^2 - 1 = 4a^4 + 4a^2 = (2a^2)^2 + (2a)^2.$$
