A308453 Number of integer-sided triangles with perimeter n whose smallest side length is squarefree.

0, 0, 1, 0, 1, 1, 2, 1, 3, 2, 4, 2, 4, 2, 5, 3, 6, 5, 8, 6, 10, 8, 12, 9, 13, 10, 14, 10, 14, 11, 15, 12, 17, 14, 19, 16, 21, 18, 24, 20, 26, 23, 29, 25, 32, 28, 35, 31, 38, 34, 42, 37, 45, 40, 48, 42, 51, 45, 54, 47, 56, 49, 59, 52, 62, 56, 66, 59, 70, 63

Offset

1,7

Links

Table of n, a(n) for n=1..70.

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

Example

There exist A005044(12) = 3 integer-sided triangles with perimeter = 12; these three triangles have respectively sides: (2, 5, 5), (3, 4, 5) or (4, 4, 4). Only the last one: (4, 4, 4) has a smallest side length = 4 that is not squarefree, so a(12) = 2. - Bernard Schott, Jan 22 2023

Mathematica

Table[Sum[Sum[ MoebiusMu[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, A308454.

Sequence in context: A355140 A393053 A286632 * A286585 A028914 A204539

Adjacent sequences: A308450 A308451 A308452 * A308454 A308455 A308456

Keyword

nonn

Author

Wesley Ivan Hurt, May 27 2019

Status

approved