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A385737 Perimeters of nondegenerate triangles with integer areas, whose side lengths are triangular numbers.

%I A385737 #7 Jul 21 2025 17:37:40
%S A385737 176,224,264,336,504,644,756,950,1196,1232,1280,1500,1566,1650,1700,
%T A385737 2100,2112,2250,2366,2754,3036,3306,5676,5796,7296,8064,8316,8526,
%U A385737 9576,10206,10260,12474,13200,15872,16236,16896,17094,17150,20172,21714,21726,22382,22644
%N A385737 Perimeters of nondegenerate triangles with integer areas, whose side lengths are triangular numbers.
%C A385737 224 and 1280 are the only perimeters <= 10^6 of nondegenerate triangles whose side lengths (28, 91, 105 or 325, 325, 630, respectively) and areas (1176 or 25200, respectively) are triangular numbers.
%H A385737 Felix Huber, <a href="/A385737/b385737.txt">Table of n, a(n) for n = 1..315</a>
%H A385737 Felix Huber, <a href="/A385737/a385737.txt">Maple program to compute the triangles (incl. areas) with perimeter a(n)</a>
%H A385737 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularNumber.html">Triangular Number</a>
%e A385737 176 is a term because it is the perimeter of the triangle [55, 55, 66], where 55 and 66 are triangular numbers, which has an integer area of sqrt(88*(88 - 55)*(88 - 55)*(88 - 66)) = 1452.
%e A385737 224 is a term because it is the perimeter of the triangle [28, 91, 105], where 28, 91 and 105 are triangular numbers, which has an integer area of sqrt(112*(112 - 28)*(112 - 91)*(112 - 105)) = 1176 (which is also a triangular number).
%p A385737 A385737:=proc(P) # To get all perimeters <= P.
%p A385737     local p,x,y,z,u,v,w,s;
%p A385737     p:=[];
%p A385737     for z to floor((sqrt(24*P+9)-3)/6) do
%p A385737         for x from z to floor((sqrt(4*P-3)-1)/2) do
%p A385737             for y from max(z,floor((sqrt(1+4*(x^2+x-z^2-z))-1)/2)+1) to min(x,floor((sqrt(1+4*(2*P-x^2-x-z^2-z))-1)/2)) do
%p A385737             	u:=z*(z+1)/2;
%p A385737             	v:=y*(y+1)/2;
%p A385737             	w:=x*(x+1)/2;
%p A385737             	s:=(u+v+w)/2;
%p A385737             	if issqr(s*(s-u)*(s-v)*(s-w)) then
%p A385737                	    p:=[op(p),u+v+w]
%p A385737                	fi
%p A385737             od
%p A385737         od
%p A385737     od;
%p A385737     return op(sort(p))
%p A385737 end proc;
%p A385737 A385737(22644);
%Y A385737 Subsequence of A380875.
%Y A385737 Cf. A000217, A051516, A070083, A385736, A385872.
%K A385737 nonn
%O A385737 1,1
%A A385737 _Felix Huber_, Jul 16 2025