A052344 Number of ways to write n as the unordered sum of two nonzero triangular numbers.

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, 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, 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

Offset: 0

Author

Christian G. Bower, Jan 23 2000

Keywords

• nonn

Comments

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

Number of partitions of 2n into two promic numbers > 1. - Wesley Ivan Hurt, Jun 09 2021

Links

• T. D. Noe, Table of n, a(n) for n = 0..10000

Crossrefs

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

Column k=2 of A319797.

Cf. A005369, A010054.

Programs

Maple

G:= (1/8)*(JacobiTheta2(0, sqrt(q))^2-4*JacobiTheta2(0, sqrt(q))*q^(1/8)+2*JacobiTheta2(0, q))/q^(1/4):
S:= series(G,q,1001):
seq(coeff(S,q,j),j=0..1000); # Robert Israel, Feb 24 2016

Mathematica

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]

Formula

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

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

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