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%I #8 Jun 18 2020 13:57:37
%S 0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,20,21,22,0,24,50,26,27,56,29,
%T 30,0,32,66,34,0,108,37,76,39,120,82,126,43,220,90,92,47,192,98,150,
%U 102,364,212,270,220,392,399,348,354,600,427,310,378,512,455
%N Sum of the perimeters of all integer-sided triangles with perimeter n whose side lengths are nonsquarefree.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Integer_triangle">Integer Triangle</a>
%F a(n) = n * A308424(n).
%F a(n) = n * Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * (1 - mu(i)^2) * (1 - mu(k)^2) * (1 - mu(n-i-k)^2), where mu is the Möbius function (A008683).
%t Table[n*Sum[Sum[(1 - MoebiusMu[i]^2)*(1 - MoebiusMu[k]^2)*(1 - MoebiusMu[n - i - k]^2)*Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]
%Y Cf. A008683, A308424.
%K nonn
%O 1,12
%A _Wesley Ivan Hurt_, May 26 2019