A307987 Number of integer-sided triangles with perimeter n and at least one nonsquarefree side.

0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2, 2, 5, 5, 8, 7, 9, 7, 8, 8, 13, 12, 13, 12, 14, 12, 13, 13, 19, 18, 19, 19, 24, 24, 24, 24, 30, 31, 31, 31, 32, 32, 32, 33, 39, 40, 46, 46, 49, 47, 55, 54, 63, 60, 64, 63, 67, 62, 66, 66, 75, 72, 76, 77, 79, 75, 79

Offset

1,9

Links

Table of n, a(n) for n=1..71.

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

a(n) = A005044(n) - A308061(n).

Mathematica

Table[Sum[Sum[(1 - 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}]

Crossrefs

Cf. A005044, A008683, A308061.

Sequence in context: A008857 A244463 A307590 * A047935 A365882 A103795

Adjacent sequences: A307984 A307985 A307986 * A307988 A307989 A307990

Keyword

nonn

Author

Wesley Ivan Hurt, May 18 2019

Status

approved