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A385872 Areas of nondegenerate triangles with perimeter A385737(n) whose side lengths are triangular numbers.

%I A385872 #12 Jul 22 2025 22:30:17
%S A385872 1452,1176,2376,3780,8316,10626,14742,28500,12558,32340,25200,94500,
%T A385872 18792,130680,89250,158760,130680,155250,53508,93636,122958,208278,
%U A385872 893970,1199772,2183328,1130976,2058210,1414098,3160080,4000752,3898800,324324,4900500,1845120,7427970
%N A385872 Areas of nondegenerate triangles with perimeter A385737(n) whose side lengths are triangular numbers.
%C A385872 45189144 is the smallest integer area of a right triangle whose sides are triangular numbers. This area corresponds to the triangle [8778, 10296, 13530].
%C A385872 From _David A. Corneth_, Jul 18 2025: (Start) If sidelengths are u, v, w where 0 < u < v < w < u + v then the area can be written as A = ((u + v + w) * (u + v - w) * (u - v + w) * (-u + v + w)) / 16 = k^2. If A is a square then 16*A is a square (possible extraneous resulting from this can be removed at the end).
%C A385872 We may rewrite 16*A as ((u + v)^2 - w^2) * (w^2 - (v - u)^2) = k^2
%C A385872 Since their product is a square we may write
%C A385872 ((u + v)^2 - w^2) * t^2 = (w^2 - (v - u)^2). where t > 1 is a rational. When u, v and t are chosen we can solve for w.
%C A385872 w^2 = (t^2*(u-v)^2 + (u+v)^2) / (t^2 + 1). (End)
%H A385872 Felix Huber, <a href="/A385872/b385872.txt">Table of n, a(n) for n = 1..315</a>
%H A385872 Felix Huber, <a href="/A385872/a385872.txt">Maple program</a>
%H A385872 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularNumber.html">Triangular Number</a>
%e A385872 a(1) = 1452 is the area of the triangle [55, 55, 66] with perimeter A385737(1) = 176, where 55 and 66 are triangular numbers.
%e A385872 a(2) = 1176 is the area of the triangle [28, 91, 105] with perimeter A385737(2) = 224, where 28, 91 and 105 are triangular numbers.
%e A385872 From _David A. Corneth_, Jul 18 2025: (Start)
%e A385872 From (u, v) = (28, 91) we get
%e A385872 ((u + v)^2 - w^2) * t^2 = (w^2 - (v - u)^2)
%e A385872 (119 - w^2) * t^2 = (w^2 - 63^2). Testing t = 2/3 gives the desired w. (End)
%p A385872 A385872:=proc(P) # To get all integer areas of triangles with perimeters <= P.
%p A385872     local p,x,y,z,u,v,w,s,i;
%p A385872     p:=[];
%p A385872     for z to floor((sqrt(24*P+9)-3)/6) do
%p A385872         for x from z to floor((sqrt(4*P-3)-1)/2) do
%p A385872             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 A385872             	u:=z*(z+1)/2;
%p A385872             	v:=y*(y+1)/2;
%p A385872             	w:=x*(x+1)/2;
%p A385872             	s:=(u+v+w)/2;
%p A385872             	if issqr(s*(s-u)*(s-v)*(s-w)) then
%p A385872                	    p:=[op(p),[u+v+w,sqrt(s*(s-u)*(s-v)*(s-w))]]
%p A385872                	fi
%p A385872             od
%p A385872         od
%p A385872     od;
%p A385872     return seq(sort(p)[i,2],i=1..nops(p))
%p A385872 end proc;
%p A385872 A385872(16236);
%Y A385872 Cf. A000217, A051516, A188158, A385736, A385737.
%K A385872 nonn
%O A385872 1,1
%A A385872 _Felix Huber_, Jul 18 2025