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

1,13

Links

Table of n, a(n) for n=1..82.

Wikipedia, Integer Triangle

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).

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}]

Crossrefs

Cf. A008683, A070093.

Sequence in context: A075256 A334523 A001480 * A270755 A305030 A110917

Adjacent sequences: A308114 A308115 A308116 * A308118 A308119 A308120

Keyword

nonn

Author

Wesley Ivan Hurt, May 13 2019

Status

approved