A335621 Number of integer-sided triangles with perimeter n such that the sum of each pair of side lengths is squarefree.

0, 0, 1, 0, 0, 0, 0, 1, 2, 1, 1, 1, 0, 0, 1, 1, 2, 3, 4, 2, 3, 2, 2, 4, 5, 4, 4, 3, 3, 3, 3, 5, 5, 4, 4, 6, 8, 5, 8, 8, 12, 7, 12, 7, 10, 6, 10, 10, 15, 14, 20, 14, 18, 17, 21, 20, 25, 20, 23, 18, 19, 16, 20, 22, 24, 21, 25, 21, 22, 20, 22, 23, 28, 22, 28, 22, 24, 20, 23, 25

Offset: 1

Author

Wesley Ivan Hurt, Oct 02 2020

Keywords

• nonn

Links

• Wikipedia, Integer Triangle

Crossrefs

Cf. A008683, A308061 (each side length is squarefree).

Cf. A335621 (averages of each pair of side lengths is prime).

Programs

Mathematica

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