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A308097 Take the integer-sided triangles with perimeter n and integer area. Then a(n) is the sum of the areas of all the triangles and the squares on their sides.

%I A308097 #10 Feb 01 2021 21:27:36
%S A308097 0,0,0,0,0,0,0,0,0,0,0,56,0,0,0,98,0,126,0,0,0,0,0,224,0,0,0,0,0,368,
%T A308097 0,826,0,0,0,2012,0,0,0,638,0,1390,0,756,0,0,0,2692,0,1928,0,0,0,4764,
%U A308097 0,1334,0,0,0,4434,0,0,0,8354,0,1778,0,1794,0,3800,0
%N A308097 Take the integer-sided triangles with perimeter n and integer area. Then a(n) is the sum of the areas of all the triangles and the squares on their sides.
%H A308097 Wikipedia, <a href="https://en.wikipedia.org/wiki/Integer_triangle">Integer Triangle</a>
%F A308097 a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} (i^2 + k^2 + (n-i-k)^2) * m * (1 - ceiling(m) + floor(m)) * sign(floor((i+k)/(n-i-k+1))), where m = sqrt((n/2)*(n/2-i)*(n/2-k)*(i+k-n/2)).
%t A308097 Table[Sum[Sum[(i^2 + k^2 + (n - i - k)^2 + Sqrt[(n/2) (n/2 - i) (n/2 - k) (i + k - n/2)]) (1 - Ceiling[Sqrt[(n/2) (n/2 - i) (n/2 - k) (i + k - n/2)]] + Floor[Sqrt[(n/2) (n/2 - i) (n/2 - k) (i + k - n/2)]])*Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]
%Y A308097 Cf. A308091.
%K A308097 nonn
%O A308097 1,12
%A A308097 _Wesley Ivan Hurt_, May 12 2019