A307987 Number of integer-sided triangles with perimeter n and at least one nonsquarefree side.

0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2, 2, 5, 5, 8, 7, 9, 7, 8, 8, 13, 12, 13, 12, 14, 12, 13, 13, 19, 18, 19, 19, 24, 24, 24, 24, 30, 31, 31, 31, 32, 32, 32, 33, 39, 40, 46, 46, 49, 47, 55, 54, 63, 60, 64, 63, 67, 62, 66, 66, 75, 72, 76, 77, 79, 75, 79

Offset: 1

Author

Wesley Ivan Hurt, May 18 2019

Keywords

• nonn

Links

• Wikipedia, Integer Triangle

Crossrefs

Cf. A005044, A008683, A308061.

Programs

Mathematica

Table[Sum[Sum[(1 - MoebiusMu[i]^2*MoebiusMu[k]^2*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 * mu(k)^2 * mu(n-i-k)^2), where mu is the Möbius function (A008683).

a(n) = A005044(n) - A308061(n).