A308424 Number of integer-sided triangles with perimeter n whose side lengths are nonsquarefree.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 2, 1, 1, 2, 1, 1, 0, 1, 2, 1, 0, 3, 1, 2, 1, 3, 2, 3, 1, 5, 2, 2, 1, 4, 2, 3, 2, 7, 4, 5, 4, 7, 7, 6, 6, 10, 7, 5, 6, 8, 7, 6, 5, 12, 5, 5, 4, 9, 7, 5, 6, 11, 6, 4, 5, 9, 8, 5, 5, 13, 5

Offset: 1

Author

Wesley Ivan Hurt, May 26 2019

Keywords

• nonn

Links

• Wikipedia, Integer Triangle

Crossrefs

Cf. A008683, A308061.

Programs

Mathematica

Table[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}]

Formula

a(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).