A385736 -id:A385736 - cp's OEIS frontend

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

176, 224, 264, 336, 504, 644, 756, 950, 1196, 1232, 1280, 1500, 1566, 1650, 1700, 2100, 2112, 2250, 2366, 2754, 3036, 3306, 5676, 5796, 7296, 8064, 8316, 8526, 9576, 10206, 10260, 12474, 13200, 15872, 16236, 16896, 17094, 17150, 20172, 21714, 21726, 22382, 22644

Offset: 1

Felix Huber, Jul 16 2025

nonn

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.

			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.
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).

Felix Huber, Table of n, a(n) for n = 1..315
Felix Huber, Maple program to compute the triangles (incl. areas) with perimeter a(n)
Eric Weisstein's World of Mathematics, Triangular Number

Subsequence of A380875.

Cf. A000217, A051516, A070083, A385736, A385872.

A385737:=proc(P) # To get all perimeters <= P.
    local p,x,y,z,u,v,w,s;
    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]
               	fi
            od
        od
    od;
    return op(sort(p))
end proc;
A385737(22644);

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

Original entry on oeis.org

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

Felix Huber, Jul 18 2025

nonn

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)

			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)

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

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

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);

A385860 a(n) is the number of distinct multisets of sides of quadrilaterals with perimeter n, where all four sides are squares.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 3, 0, 1, 1, 0, 1, 2, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 0, 2, 1, 1, 1

Offset: 0

Felix Huber, Jul 22 2025

nonn

a(n) is the number of partitions of n into 4 nonzero squares < n/2.

			The a(51) = 1 multiset is [1, 9, 16, 25].
The a(52) = 3 multisets are [1, 1, 25, 25], [4, 16, 16, 16] and [9, 9, 9, 25].

Felix Huber, Table of n, a(n) for n = 0..10000

Cf. A000290, A000414, A000534, A025428, A062890, A385736.

# After Alois P. Heinz (A025428)
b:=proc(n,i,t)
    option remember;
    `if`(n=0,`if`(t=0,1,0),`if`(i<1 or t<1, 0, b(n,i-1,t)+`if`(i^2>n,0,b(n-i^2,i,t-1))))
    end:
A385860:=n->b(n,floor(sqrt((n-1)/2)),4):
seq(A385860(n),n=0..87);

a(n) <= A025428(n).

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

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

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Programs

Maple

A385860 a(n) is the number of distinct multisets of sides of quadrilaterals with perimeter n, where all four sides are squares.

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Keywords

Comments

Examples

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Crossrefs

Programs

Maple

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