%I #17 Jan 06 2024 01:24:20

%S 0,0,1,0,1,0,1,1,0,1,0,1,1,1,0,0,2,0,1,0,1,1,1,0,1,1,0,1,0,1,1,2,0,0,

%T 1,0,1,1,1,1,0,0,2,1,0,0,2,0,1,1,0,2,0,0,0,1,2,1,1,0,1,1,0,0,1,1,1,1,

%U 0,1,1,0,2,1,0,0,2,0,0,1,0,3,0,1,1,0,0,1,1,0,1,1,1,1,2,0,0,1,0,1,1,1

%N Number of ways to write n as the unordered sum of two nonzero triangular numbers.

%C Number of ways to write 8*n+2 as the unordered sum of two odd squares > 1. - _Robert Israel_, Feb 24 2016

%C Number of partitions of 2n into two promic numbers > 1. - _Wesley Ivan Hurt_, Jun 09 2021

%H T. D. Noe, <a href="/A052344/b052344.txt">Table of n, a(n) for n = 0..10000</a>

%F G.f.: (Theta_2(sqrt(x))^2 - 4*x^(1/8)*Theta_2(sqrt(x)) + 2*Theta_2(x))/(8*x^(1/4)) where Theta_2 is a Jacobi theta function. - _Robert Israel_, Feb 24 2016

%F a(n) = Sum_{k=1..n} c(k) * c(2*n-k), where c(n) is the characteristic function of promic numbers (A005369). - _Wesley Ivan Hurt_, Jun 09 2021

%F a(n) = Sum_{k=1..floor(n/2)} c(k) * c(n-k), where c = A010054. - _Wesley Ivan Hurt_, Jan 06 2024

%p G:= (1/8)*(JacobiTheta2(0, sqrt(q))^2-4*JacobiTheta2(0, sqrt(q))*q^(1/8)+2*JacobiTheta2(0, q))/q^(1/4):

%p S:= series(G,q,1001):

%p seq(coeff(S,q,j),j=0..1000); # _Robert Israel_, Feb 24 2016

%t nn=150; tri=Accumulate[Range[nn]]; t=Table[0, {tri[[-1]]}]; Do[n=tri[[i]]+tri[[j]]; If[n <= tri[[-1]], t[[n]]++], {i,nn}, {j,i}]; t=Prepend[t,0]

%Y Cf. A000217, A052343-A052348, A053587, A056303, A056304.

%Y Column k=2 of A319797.

%Y Cf. A005369, A010054.

%K nonn

%O 0,17

%A _Christian G. Bower_, Jan 23 2000