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%I #8 Jun 16 2020 13:49:38
%S 0,0,1,0,0,0,0,0,1,0,0,4,0,0,0,0,0,0,1,0,0,4,0,0,0,0,9,0,0,0,0,0,1,9,
%T 0,4,0,0,0,0,9,0,0,0,0,0,0,16,0,0,1,0,0,4,0,0,16,0,9,0,0,0,0,0,0,16,0,
%U 0,0,0,0,0,1,0,25,4,16,0,0,0,9,0,0,0,0
%N Sum of the smallest side lengths of all integer-sided triangles with perimeter n whose sides are square numbers.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Integer_triangle">Integer Triangle</a>
%F a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * A010052(i) * A010052(k) * A010052(n-i-k) * k.
%t Table[Sum[Sum[k (Floor[Sqrt[i]] - Floor[Sqrt[i - 1]]) (Floor[Sqrt[k]] - Floor[Sqrt[k - 1]]) (Floor[Sqrt[n - k - i]] - Floor[Sqrt[n - k - i - 1]])*Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]
%Y Cf. A010052, A308275.
%K nonn
%O 1,12
%A _Wesley Ivan Hurt_, May 18 2019