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How many ordered $2$ tuples can be chosen from $(1,2,..n)$ such that the sum of the two numbers within the tuple is a perfect square?

Will generating functions help here?

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    $\begingroup$ You can arrange the ordered 2-tuples into a square array. The 2-tuples with constant sum appear along diagonals. So for squares in $[0,n]$ the sequence of counts of 2-tuples increases, then for squares in $[n,2n]$, the sequence of counts of 2-tuples decreases. Is there something about this that is blocking your progress? $\endgroup$ – Eric Towers Apr 16 '16 at 8:30

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