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%I #10 Oct 03 2023 17:41:31
%S 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,
%T 23,16,25,19,27,15,31,23,31,21,36,21,40,28,43,33,47,26,50,36,52,38,60,
%U 36,62,45,66,52,72,36,75,58,74,56,84,49,91,64,93,65
%N Number of integer-sided triangles with perimeter n and at least one pair of side lengths that are not coprime.
%H Robert Israel, <a href="/A308308/b308308.txt">Table of n, a(n) for n = 1..2500</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Integer_triangle">Integer Triangle</a>
%F 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.
%F a(n) = A005044(n) - A308074(n).
%p N:= 100: # for a(1)..a(N)
%p A:= Vector(N):
%p for a from 1 to floor(N/3) do
%p for b from a to floor((N-a)/2) do
%p if igcd(a,b) = 1 then
%p C:= select(c -> igcd(c,a*b) <> 1, [$b .. min(a+b-1,N-a-b)])+~ (a+b)
%p else C:= [$b .. min(a+b-1,N-a-b)] +~ (a+b)
%p fi;
%p A[C]:= A[C] +~ 1
%p od od:
%p convert(A,list); # _Robert Israel_, Oct 03 2023
%t 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}]
%Y Cf. A005044, A308074.
%K nonn,look
%O 1,7
%A _Wesley Ivan Hurt_, May 19 2019
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