A385872 -id:A385872 - cp's OEIS frontend

A385736 a(n) is the number of distinct nondegenerate triangles with perimeter n whose side lengths are triangular numbers.

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

Offset: 0

Felix Huber, Jul 16 2025

0, 1, 6, 10, 28, 55 are the only triangular numbers <= 10^6 that are not perimeters of triangles whose side lengths are triangular numbers. Conjecture: There are no other triangular numbers that have this property.

			The a(31) = 2 distinct nondegenerate triangles with perimeter 31 and whose side lengths are triangular numbers are [1, 15, 15] and [6, 10, 15].

Felix Huber, Table of n, a(n) for n = 0..10000
Felix Huber, Maple program to compute the counted triangles
Eric Weisstein's World of Mathematics, Triangular Number

Cf. A000217, A005044, A051516, A070088, A366398, A380875, A385737, A385860, A385872.

A385736:=proc(N) # To get the first N + 1 terms.
    local p,x,y,z,i;
    p:=[];
    for z to floor((sqrt(24*N+9)-3)/6) do
        for x from z to floor((sqrt(4*N-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*N-x^2-x-z^2-z))-1)/2)) do
                p:=[op(p),z*(z+1)/2+y*(y+1)/2+x*(x+1)/2]
            od
        od
    od;
    return seq(numboccur(p,i),i=0..N)
end proc;
A385736(87);

Trivial upper bound: a(n) <= A005044(n).

a(A385737(n)) >= 1.

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

Original entry on oeis.org

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);

cp's OEIS Frontend

A385736 a(n) is the number of distinct nondegenerate triangles with perimeter n whose side lengths are triangular numbers.

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