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

1452, 1176, 2376, 3780, 8316, 10626, 14742, 28500, 12558, 32340, 25200, 94500, 18792, 130680, 89250, 158760, 130680, 155250, 53508, 93636, 122958, 208278, 893970, 1199772, 2183328, 1130976, 2058210, 1414098, 3160080, 4000752, 3898800, 324324, 4900500, 1845120, 7427970

Offset: 1

Author

Felix Huber, Jul 18 2025

Keywords

• nonn

Comments

45189144 is the smallest integer area of a right triangle whose sides are triangular numbers. This area corresponds to the triangle [8778, 10296, 13530].

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).

We may rewrite 16*A as ((u + v)^2 - w^2) * (w^2 - (v - u)^2) = k^2

Since their product is a square we may write

((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.

w^2 = (t^2*(u-v)^2 + (u+v)^2) / (t^2 + 1). (End)

Examples

			a(1) = 1452 is the area of the triangle [55, 55, 66] with perimeter A385737(1) = 176, where 55 and 66 are triangular numbers.
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.
From _David A. Corneth_, Jul 18 2025: (Start)
From (u, v) = (28, 91) we get
((u + v)^2 - w^2) * t^2 = (w^2 - (v - u)^2)
(119 - w^2) * t^2 = (w^2 - 63^2). Testing t = 2/3 gives the desired w. (End)
		

Links

• Felix Huber, Table of n, a(n) for n = 1..315
• Felix Huber, Maple program
• Eric Weisstein's World of Mathematics, Triangular Number

Crossrefs

Cf. A000217, A051516, A188158, A385736, A385737.

Programs

Maple

A385872:=proc(P) # To get all integer areas of triangles with perimeters <= P.
    local p,x,y,z,u,v,w,s,i;
    p:=[];
    for z to floor((sqrt(24*P+9)-3)/6) do
        for x from z to floor((sqrt(4*P-3)-1)/2) do
            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
            	u:=z*(z+1)/2;
            	v:=y*(y+1)/2;
            	w:=x*(x+1)/2;
            	s:=(u+v+w)/2;
            	if issqr(s*(s-u)*(s-v)*(s-w)) then
               	    p:=[op(p),[u+v+w,sqrt(s*(s-u)*(s-v)*(s-w))]]
               	fi
            od
        od
    od;
    return seq(sort(p)[i,2],i=1..nops(p))
end proc;
A385872(16236);