%I #21 Jun 16 2020 14:33:37
%S 0,0,0,0,0,0,0,0,52,0,76,94,212,126,426,328,724,624,1130,1020,1938,
%T 1540,2648,2568,3910,3432,5482,4970,7364,6850,9616,9072,12954,11696,
%U 16086,15576,20544,19152,25698,24240,31530,30072,38148,36630,46870,44022,55240
%N Take apart the sides of each of the integer-sided scalene 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-1)/3)} Sum_{i=k+1..floor((n-k-1)/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^9*(26 + 52*x + 90*x^2 + 97*x^3 + 94*x^4 + 71*x^5 + 56*x^6 + 31*x^7 + 17*x^8 + 5*x^9 + x^10) / ((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)
%e There is one integer-sided scalene triangle with perimeter 9: (2,3,4). The surface area of the enclosed rectangular prism is 2*(2*3 + 2*4 + 3*4) = 52. So a(9) = 52.
%t 2*Sum[Sum[(i*k + i*(n - i - k) + k*(n - i - k))*Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k + 1, Floor[(n - k - 1)/2]}], {k, Floor[(n - 1)/3]}], {n, 100}]
%Y Cf. A308233.
%K nonn
%O 1,9
%A _Wesley Ivan Hurt_, May 16 2019