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A308308
Number of integer-sided triangles with perimeter n and at least one pair of side lengths that are not coprime.
1
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
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
Sequence in context: A130722 A308307 A147541 * A024162 A334677 A365876
KEYWORD
nonn,look
AUTHOR
Wesley Ivan Hurt, May 19 2019
STATUS
approved