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%I #33 Oct 20 2021 15:45:20
%S 0,0,1,0,1,2,3,2,6,5,9,9,13,13,22,18,27,29,38,35,51,48,64,63,79,78,
%T 103,95,120,122,147,141,177,171,207,204,240,237,286,273,322,323,372,
%U 362,426,416,480,474,538,532,613,594,675,674,755,740,840,825,925
%N Sum of the smallest side lengths of all integer-sided triangles with perimeter n.
%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))) * k.
%F Conjectures from _Colin Barker_, May 12 2019: (Start)
%F G.f.: x^3*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6) / ((1 - x)^4*(1 + x)^3*(1 + x^2)^2*(1 + x + x^2)^2).
%F a(n) = -a(n-1) + 2*a(n-3) + 4*a(n-4) + 2*a(n-5) - a(n-6) - 5*a(n-7) - 5*a(n-8) - a(n-9) + 2*a(n-10) + 4*a(n-11) + 2*a(n-12) - a(n-14) - a(n-15) for n>15.
%F (End)
%t Table[Sum[Sum[k*Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]
%Y Cf. A005044.
%K nonn
%O 1,6
%A _Wesley Ivan Hurt_, May 11 2019