%I #15 Jun 18 2020 13:50:12
%S 0,0,6,0,16,24,62,42,154,130,304,280,530,504,990,820,1448,1452,2260,
%T 2040,3318,3080,4634,4398,6256,6006,8674,7952,11046,10840,14424,13608,
%U 18402,17544,22980,22128,28248,27360,35208,33330,42040,41202,50864,48840,60796
%N Take apart the sides of each of the integer-sided triangles with perimeter n (at their vertices) and rearrange them orthogonally in 3-space so that their endpoints coincide at a single point. a(n) is the total surface area of all rectangular prisms enclosed in this way.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Integer_triangle">Integer Triangle</a>
%F a(n) = 2 * Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * (i*k + i*(n-i-k) + k*(n-i-k)).
%F Conjectures from _Colin Barker_, May 16 2019: (Start)
%F G.f.: 2*x^3*(3 + 6*x + 14*x^2 + 25*x^3 + 50*x^4 + 69*x^5 + 92*x^6 + 81*x^7 + 73*x^8 + 53*x^9 + 41*x^10 + 22*x^11 + 11*x^12) / ((1 - x)^5*(1 + x)^4*(1 + x^2)^3*(1 + x + x^2)^3).
%F a(n) = -2*a(n-1) - 2*a(n-2) + a(n-3) + 7*a(n-4) + 10*a(n-5) + 7*a(n-6) - 5*a(n-7) - 17*a(n-8) - 19*a(n-9) - 9*a(n-10) + 9*a(n-11) + 19*a(n-12) + 17*a(n-13) + 5*a(n-14) - 7*a(n-15) - 10*a(n-16) - 7*a(n-17) - a(n-18) + 2*a(n-19) + 2*a(n-20) + a(n-21) for n>21.
%F (End)
%t Table[2*Sum[Sum[(i*k + i*(n - i - k) + k*(n - i - k))*Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]
%Y Cf. A308235.
%K nonn
%O 1,3
%A _Wesley Ivan Hurt_, May 16 2019