A308308 Number of integer-sided triangles with perimeter n and at least one pair of side lengths that are not coprime.

0, 0, 0, 0, 1, 1, 2, 1, 3, 2, 4, 2, 5, 4, 6, 4, 8, 5, 10, 6, 11, 9, 13, 7, 15, 12, 16, 11, 19, 11, 23, 16, 25, 19, 27, 15, 31, 23, 31, 21, 36, 21, 40, 28, 43, 33, 47, 26, 50, 36, 52, 38, 60, 36, 62, 45, 66, 52, 72, 36, 75, 58, 74, 56, 84, 49, 91, 64, 93, 65

Offset

1,7

Links

Robert Israel, Table of n, a(n) for n = 1..2500

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))) * (1 - [gcd(i,k) * gcd(i,n-i-k) * gcd(k,n-i-k) = 1]), where [] is the Iverson bracket.

a(n) = A005044(n) - A308074(n).

Maple

N:= 100: # for a(1)..a(N)
A:= Vector(N):
for a from 1 to floor(N/3) do
  for b from a to floor((N-a)/2) do
    if igcd(a, b) = 1 then
      C:= select(c -> igcd(c, a*b) <> 1, [$b .. min(a+b-1, N-a-b)])+~ (a+b)
    else C:= [$b .. min(a+b-1, N-a-b)] +~ (a+b)
    fi;
    A[C]:= A[C] +~ 1
od od:
convert(A, list); # Robert Israel, Oct 03 2023

Mathematica

Table[Sum[Sum[(1 - KroneckerDelta[GCD[i, k]*GCD[i, n - i - k]*GCD[k, n - i - k], 1])*Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]

Crossrefs

Cf. A005044, A308074.

Sequence in context: A130722 A308307 A147541 * A024162 A334677 A365876

Adjacent sequences: A308305 A308306 A308307 * A308309 A308310 A308311

Keyword

nonn,look

Author

Wesley Ivan Hurt, May 19 2019

Status

approved