A112603 Number of representations of n as the sum of a square and a triangular number.

1, 3, 2, 1, 4, 2, 1, 4, 0, 2, 5, 2, 2, 0, 2, 3, 4, 2, 0, 6, 0, 1, 4, 0, 2, 4, 4, 0, 3, 2, 2, 4, 2, 0, 0, 2, 3, 8, 0, 2, 4, 0, 2, 0, 2, 3, 6, 0, 0, 4, 2, 2, 4, 2, 2, 3, 2, 2, 0, 4, 0, 4, 0, 0, 8, 2, 1, 4, 0, 0, 8, 2, 2, 0, 2, 2, 0, 2, 1, 4, 2, 4, 6, 0, 2, 4, 0, 4, 0, 0, 0, 7, 4, 0, 4, 2, 2, 0, 0, 0, 6, 2, 4, 4, 2

Offset

0,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

a(n) = A002325(8n+1). [Hirschhorn]

Expansion of q^(-1/8) * eta(q^2)^7 / (eta(q)^3 * eta(q^4)^2) in powers of q. - Michael Somos, Sep 29 2006

Expansion of phi(q) * psi(q) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Sep 29 2006

Euler transform of period 4 sequence [ 3, -4, 3, -2, ...]. - Michael Somos, Sep 29 2006

G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 2^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A139093. - Michael Somos, Mar 16 2011

G.f.: (Sum_{k} x^(k^2)) * (Sum_{k>0} x^((k^2 - k)/2)). - Michael Somos, Sep 29 2006

Example

a(4) = 4 since we can write 4 = 2^2 + 0 = (-2)^2 + 0 = 1^2 + 3 = (-1)^2 + 3.
1 + 3*x + 2*x^2 + x^3 + 4*x^4 + 2*x^5 + x^6 + 4*x^7 + 2*x^9 + 5*x^10 + ...
q + 3*q^9 + 2*q^17 + q^25 + 4*q^33 + 2*q^41 + q^49 + 4*q^57 + 2*q^73 + ...

Mathematica

a[n_] := DivisorSum[8n + 1, KroneckerSymbol[-2, #]&]; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Dec 06 2015, adapted from PARI *)

Prog

(PARI) {a(n) = if( n<0, 0, n = 8*n + 1; sumdiv( n, d, kronecker( -2, d)))} /* Michael Somos, Sep 29 2006 */
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 /(eta(x + A)^3 * eta(x^4 + A)^2), n))} /* Michael Somos, Sep 29 2006 */

Crossrefs

Cf. A139093.

Sequence in context: A194520 A082727 A264597 * A097294 A060848 A265271

Adjacent sequences: A112600 A112601 A112602 * A112604 A112605 A112606

Keyword

nonn

Author

James A. Sellers, Dec 21 2005

Status

approved