A308061 Number of integer-sided triangles with perimeter n whose side lengths are squarefree.

0, 0, 1, 0, 1, 1, 2, 1, 1, 0, 2, 1, 3, 2, 5, 3, 3, 2, 2, 1, 3, 3, 6, 4, 3, 2, 6, 4, 7, 7, 11, 8, 8, 6, 11, 8, 9, 6, 13, 9, 10, 6, 13, 9, 16, 12, 20, 15, 17, 12, 15, 10, 16, 14, 15, 11, 12, 10, 16, 12, 18, 18, 25, 19, 21, 19, 26, 19, 29, 27, 35, 27, 29, 27

Offset: 1

Author

Wesley Ivan Hurt, May 10 2019

Keywords

• nonn
• easy

Links

• Wikipedia, Integer Triangle

Crossrefs

Cf. A005044, A008683.

Programs

Mathematica

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