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

1,5

Links

Robert Israel, Table of n, a(n) for n = 1..1000

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

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}]

Crossrefs

Cf. A008683, A308116.

Sequence in context: A146316 A258803 A157495 * A308515 A361328 A128134

Adjacent sequences: A308140 A308141 A308142 * A308144 A308145 A308146

Keyword

nonn

Author

Wesley Ivan Hurt, May 14 2019

Status

approved