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

%I A385736 #13 Jul 21 2025 22:30:19
%S A385736 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,
%T A385736 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,
%U A385736 0,0,1,1,0,1,1,1,0,1,2,1,0,0,1,0,1,1,0,1
%N A385736 a(n) is the number of distinct nondegenerate triangles with perimeter n whose side lengths are triangular numbers.
%C A385736 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.
%H A385736 Felix Huber, <a href="/A385736/b385736.txt">Table of n, a(n) for n = 0..10000</a>
%H A385736 Felix Huber, <a href="/A385736/a385736.txt">Maple program to compute the counted triangles</a>
%H A385736 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularNumber.html">Triangular Number</a>
%F A385736 Trivial upper bound: a(n) <= A005044(n).
%F A385736 a(A385737(n)) >= 1.
%e A385736 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].
%p A385736 A385736:=proc(N) # To get the first N + 1 terms.
%p A385736     local p,x,y,z,i;
%p A385736     p:=[];
%p A385736     for z to floor((sqrt(24*N+9)-3)/6) do
%p A385736         for x from z to floor((sqrt(4*N-3)-1)/2) do
%p A385736             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 A385736                 p:=[op(p),z*(z+1)/2+y*(y+1)/2+x*(x+1)/2]
%p A385736             od
%p A385736         od
%p A385736     od;
%p A385736     return seq(numboccur(p,i),i=0..N)
%p A385736 end proc;
%p A385736 A385736(87);
%Y A385736 Cf. A000217, A005044, A051516, A070088, A366398, A380875, A385737, A385860, A385872.
%K A385736 nonn,easy
%O A385736 0,32
%A A385736 _Felix Huber_, Jul 16 2025