A308103 Number of scalene triangles with perimeter n whose side lengths are squarefree.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 1, 0, 1, 0, 0, 1, 1, 3, 2, 0, 1, 2, 2, 4, 5, 7, 5, 3, 4, 6, 4, 4, 5, 7, 5, 4, 5, 7, 5, 9, 11, 13, 10, 9, 10, 9, 6, 10, 11, 10, 7, 6, 8, 9, 8, 12, 16, 17, 14, 13, 16, 18, 13, 22, 24, 27, 20, 19, 23, 24, 22, 29, 34

Offset: 1

Author

Wesley Ivan Hurt, May 12 2019

Keywords

• nonn

Links

• Wikipedia, Integer Triangle

Crossrefs

Cf. A008683, A308061.

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 + 1, Floor[(n - k - 1)/2]}], {k, Floor[(n - 1)/3]}], {n, 100}]

Formula

a(n) = Sum_{k=1..floor((n-1)/3)} Sum_{i=k+1..floor((n-k-1)/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).