A308117 Number of acute integer-sided triangles with perimeter n and squarefree sides.

0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 2, 1, 3, 2, 3, 2, 2, 1, 2, 1, 2, 1, 2, 2, 4, 3, 3, 2, 5, 3, 6, 4, 6, 5, 5, 3, 6, 5, 7, 4, 6, 5, 7, 4, 8, 5, 7, 4, 8, 5, 9, 7, 10, 7, 10, 6, 9, 5, 7, 6, 10, 7, 8, 6, 7, 7, 9, 7, 12, 9, 13, 11, 14, 10, 14, 11, 15, 13, 18, 15

Offset: 1

Author

Wesley Ivan Hurt, May 13 2019

Keywords

• nonn

Links

• Wikipedia, Integer Triangle

Crossrefs

Cf. A008683, A070093.

Programs

Mathematica

Table[Sum[Sum[MoebiusMu[i]^2*MoebiusMu[k]^2*MoebiusMu[n - k - i]^2 (1 - Sign[Floor[(n - i - k)^2/(i^2 + 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)} (1-sign(floor((n-i-k)^2/(i^2+k^2)))) * sign(floor((i+k)/(n-i-k+1))) mu(i)^2 * mu(k)^2 * mu(n-i-k)^2, where mu is the Möbius function (A008683).