A308143 Take all the integer-sided triangles with perimeter n and squarefree sides a, b, and c such that a <= b <= c. a(n) is the sum of all the b's.

0, 0, 1, 0, 2, 2, 5, 3, 3, 0, 8, 5, 16, 11, 29, 18, 18, 12, 13, 7, 23, 23, 51, 35, 28, 20, 62, 44, 82, 79, 132, 98, 100, 75, 144, 108, 121, 80, 185, 131, 148, 87, 203, 145, 265, 200, 345, 264, 300, 214, 272, 187, 305, 274, 301, 216, 246, 210, 340, 258, 406

Offset: 1

Author

Wesley Ivan Hurt, May 14 2019

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 1..1000
• Wikipedia, Integer Triangle

Crossrefs

Cf. A008683, A308116.

Programs

Maple

f:= proc(n)
  local a,b,t;
  t:= 0;
  for a from 1 to n/3 do
    if not a::squarefree then next fi;
    for b from max(a, ceil((n+1)/2-a)) to (n-a)/2 do
      if b::squarefree and (n-a-b)::squarefree then t:= t+b fi
  od od;
  t
end proc:
map(f, [$1..100]); # Robert Israel, May 09 2024

Mathematica

Table[Sum[Sum[i* MoebiusMu[i]^2*MoebiusMu[k]^2*MoebiusMu[n - k - i]^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(i)^2 * mu(k)^2 * mu(n-i-k)^2 * i, where mu is the Möbius function (A008683).