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

Author

Wesley Ivan Hurt, May 27 2019

Keywords

• nonn

Examples

			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
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Wikipedia, Integer Triangle

Crossrefs

Cf. A005044, A008683, A308453.

Programs

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}]

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).