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%I #11 Jun 16 2020 14:23:47
%S 0,0,1,0,2,2,5,3,10,7,16,13,24,20,38,29,50,45,69,58,92,79,117,104,146,
%T 131,186,162,222,205,270,243,324,294,381,351,444,411,523,477,596,560,
%U 686,636,784,730,886,832,996,938,1127,1052,1250,1188,1395,1315
%N Take all the integer-sided triangles with perimeter n and sides a, b, and c such that a <= b <= c. a(n) is the sum of all the b's.
%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))) * i.
%F Conjectures from _Colin Barker_, May 17 2019: (Start)
%F G.f.: x^3*(1 + x + 2*x^2 + 2*x^3 + 3*x^4 + 2*x^5 + 2*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[i*Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]
%Y Cf. A307686 (sum of smallest sides), A307966 (sum of largest sides).
%K nonn
%O 1,5
%A _Wesley Ivan Hurt_, May 12 2019
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