A112608 Number of representations of n as a sum of a twice a square and three times a triangular number.

1, 0, 2, 1, 0, 2, 0, 0, 2, 1, 0, 4, 0, 0, 0, 0, 0, 2, 3, 0, 2, 2, 0, 0, 0, 0, 2, 2, 0, 0, 1, 0, 4, 0, 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 1, 0, 2, 2, 0, 4, 0, 0, 4, 0, 0, 0, 0, 0, 2, 0, 0, 2, 3, 0, 2, 0, 0, 2, 0, 0, 2, 2, 0, 0, 2, 0, 2, 0, 0, 2, 4, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 4, 0, 0, 2, 0, 0, 2, 4, 0, 0

Offset

0,3

Comments

The greedy inverse (first occurrence of n) starts 1, 0, 2, 18, 11, 900, 116, 44118, 515, 3105, 5702, ... - R. J. Mathar, Apr 28 2020

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

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

Formula

a(n) = d_{1, 3}(8n+3) - d_{2, 3}(8n+3) where d_{a, m}(n) equals the number of divisors of n which are congruent to a mod m.

Euler transform of period 24 sequence [0, 2, 1, -3, 0, 1, 0, -1, 1, 2, 0, -4, 0, 2, 1, -1, 0, 1, 0, -3, 1, 2, 0, -2, ...]. - Michael Somos, Jan 01 2006

Expansion of q^(-3/8)*(eta(q^4)^5*eta(q^6)^2)/(eta(q^2)^2*eta(q^3)*eta(q^8)^2) in powers of q.

a(n) = A002324(8*n+3).

Example

a(11) = 4 since we can write 11 = 2*(2)^2 + 3*1 = 2*(-2)^2 + 3*1 = 2*(1)^2 + 3*3 = 2*(-1)^2 + 3*3

Mathematica

eta[x_] := x^(1/24)*QPochhammer[x]; A112608[n_] := SeriesCoefficient[ q^(-3/8)*(eta[q^4]^5*eta[q^6]^2)/(eta[q^2]^2*eta[q^3]*eta[q^8]^2), {q, 0, n}]; Table[A112608[n], {n, 0, 50}] (* G. C. Greubel, Sep 25 2017 *)

Prog

(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^4+A)^5*eta(x^6+A)^2/ eta(x^2+A)^2/eta(x^3+A)/eta(x^8)^2, n))} /* Michael Somos, Jan 01 2006 */

Crossrefs

Sequence in context: A182033 A112214 A246962 * A058677 A262780 A033762

Adjacent sequences: A112605 A112606 A112607 * A112609 A112610 A112611

Keyword

nonn

Author

James A. Sellers, Dec 21 2005

Status

approved