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A051516 Number of triangles with perimeter n having integer sides and area.

%I A051516 #20 Feb 16 2025 08:32:41
%S A051516 0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,2,0,0,
%T A051516 0,4,0,0,0,1,0,2,0,1,0,0,0,3,0,2,0,0,0,4,0,1,0,0,0,3,0,0,0,5,0,1,0,1,
%U A051516 0,2,0,5,0,0,0,1,0,1,0,4,0,0,0,8,0,0,0,1,0,5,0,0,0,0,0,5,0,6,0,5,0,0,0,2,0,0,0,12,0,1,0
%N A051516 Number of triangles with perimeter n having integer sides and area.
%C A051516 No such triangles with odd perimeter exist.
%H A051516 Giovanni Resta, <a href="/A051516/b051516.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Seiichi Manyama)
%H A051516 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HeronianTriangle.html">Heronian Triangle</a>.
%F A051516 a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} (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)). - _Wesley Ivan Hurt_, May 11 2019
%t A051516 Table[Sum[Sum[(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}] (* _Wesley Ivan Hurt_, May 11 2019 *)
%Y A051516 Cf. A024153, A070139.
%K A051516 nonn
%O A051516 1,32
%A A051516 _David W. Wilson_