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%I #8 Feb 01 2021 19:19:08
%S 1,1,1,1,2,1,1,2,1,1,2,3,1,2,2,1,4,1,1,1,4,3,1,2,2,1,2,3,1,1,1,3,1,1,
%T 2,1,2,1,5,2,1,1,4,2,3,3,1,1,1,1,1,2,4,1,3,1,2,1,3,9,1,1,6,1,1,1,3,1,
%U 1,1,6,6,2,1,2,2,2,2,1,1,2,1,1,5,7,1,3,4,4,3,1,2,3,2
%N Number of Heronian triangles with perimeter A330869(n) whose side lengths are squarefree.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HeronianTriangle.html">Heronian Triangle</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Heronian_triangle">Heronian triangle</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Integer_triangle">Integer Triangle</a>
%F a(n) = Sum_{k=1..floor(b(n)/3)} Sum_{i=k..floor((b(n)-k)/2)} sign(floor((i+k)/(b(n)-i-k+1))) * (1-ceiling(c)+floor(c)) * mu(k)^2 * mu(i)^2 * mu(b(n)-i-k)^2, where c = sqrt((b(n)/2)*(b(n)/2-i)*(b(n)/2-k)*(b(n)/2-(b(n)-i-k))), mu is the Möbius function (A008683) and b(n) = A330869(n).
%e a(1) = 1; there is one Heronian triangle with perimeter A330869(1) = 16 whose side lengths are squarefree, which is [5,5,6].
%e a(2) = 1; there is one Heronian triangle with perimeter A330869(2) = 36 whose side lengths are squarefree, [10,13,13].
%Y Cf. A008683, A330869.
%K nonn
%O 1,5
%A _Wesley Ivan Hurt_, May 27 2020