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%I #14 Aug 15 2020 12:50:55
%S 0,0,2,0,3,4,8,5,16,12,25,22,37,33,60,47,77,74,107,93,143,127,181,167,
%T 225,209,289,257,342,327,417,384,501,465,588,555,684,648,809,750,918,
%U 883,1058,998,1210,1146,1366,1306,1534,1470,1740,1646,1925,1862,2150,2055,2390,2290,2635
%N Sum of the smallest two side lengths of all distinct 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))) * (i + k).
%F Conjectures from _Colin Barker_, Aug 10 2020: (Start)
%F G.f.: x^3*(2 + 2*x + 3*x^2 + 3*x^3 + 4*x^4 + 3*x^5 + 3*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)
%e a(3) = 2; There is one integer-sided triangle with perimeter 3, [1,1,1]. The sum of the smallest two side lengths is 1 + 1 = 2.
%e a(7) = 8; There are two distinct integer-sided triangles with perimeter 7, [1,3,3] and [2,2,3]. The sum of the smallest two side lengths of these triangles is 1 + 3 + 2 + 2 = 8.
%t Table[Sum[Sum[(i + k)*Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 80}]
%Y Cf. A005044.
%K nonn,easy
%O 1,3
%A _Wesley Ivan Hurt_, Aug 09 2020
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