A308424 Number of integer-sided triangles with perimeter n whose side lengths are nonsquarefree.

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

Offset

1,25

Links

Table of n, a(n) for n=1..85.

Wikipedia, Integer Triangle

Formula

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * (1 - mu(i)^2) * (1 - mu(k)^2) * (1 - mu(n-i-k)^2), where mu is the Möbius function (A008683).

Mathematica

Table[Sum[Sum[(1 - MoebiusMu[i]^2)*(1 - MoebiusMu[k]^2)*(1 - 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}]

Crossrefs

Cf. A008683, A308061.

Sequence in context: A375383 A143158 A336708 * A317489 A345647 A091950

Adjacent sequences: A308421 A308422 A308423 * A308425 A308426 A308427

Keyword

nonn

Author

Wesley Ivan Hurt, May 26 2019

Status

approved