A308454 Number of integer-sided triangles with perimeter n whose largest side length is squarefree.

0, 0, 1, 0, 1, 1, 2, 1, 1, 0, 3, 2, 5, 4, 7, 5, 4, 3, 2, 1, 6, 5, 10, 9, 8, 7, 13, 11, 17, 15, 21, 18, 17, 14, 22, 19, 18, 15, 24, 20, 19, 16, 26, 22, 33, 29, 40, 36, 35, 31, 30, 26, 38, 35, 33, 30, 28, 25, 38, 35, 48, 45, 58, 54, 51, 48, 62, 58, 73, 69, 84

Offset

1,7

Links

Robert Israel, Table of n, a(n) for n = 1..10000

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

Example

There exist A005044(11) = 4 integer-sided triangles with perimeter = 11; these four triangles have respectively sides: (1, 5, 5); (2, 4, 5); (3, 3, 5); (3, 4, 4). Only the last one: (3, 4, 4) has a largest side length = 4 that is not squarefree, so a(11) = 3. - Bernard Schott, Jan 24 2023

Maple

f:= proc(n) local p, v;
  v:= add(1/2*(3*p-n+1)+`if`((n-p)::even, 1/2, 0),
     p = select(numtheory:-issqrfree, [$ceil(n/3)..floor((n-1)/2)]));
end proc:
map(f, [$1..100]); # Robert Israel, Jan 16 2023

Mathematica

Table[Sum[Sum[ 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, A308453.

Sequence in context: A277488 A325794 A119513 * A085815 A088234 A228717

Adjacent sequences: A308451 A308452 A308453 * A308455 A308456 A308457

Keyword

nonn

Author

Wesley Ivan Hurt, May 27 2019

Status

approved