A112606 Number of representations of n as a sum of six times a square and a triangular number.

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

Offset

0,7

Comments

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

The greedy inverse starts 2, 0, 7, 6, 27, 300, 349, 14706, 216, 1035, 17107,... - R. J. Mathar, Apr 28 2020

References

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

Links

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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

Expansion of q^(-1/8) * eta(q^2)^2 * eta(q^12)^5 /(eta(q) * eta(q^6)^2 * eta(q^24)^2) in powers of q. - Michael Somos, Sep 29 2006

Expansion of phi(q^6) * psi(q) in powers of q where phi(), psi() are Ramanujan theta functions.

Euler transform of period 24 sequence [ 1, -1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -4, 1, -1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -2, ...]. - Michael Somos, Sep 29 2006

G.f.: (Sum_{k} x^(6*k^2)) * (Sum_{k>0} x^((k^2-k)/2)). a(3*n+2)=0. - Michael Somos, Sep 29 2006

a(n) = A123484(24*n + 3) = A112604(2*n) = A112608(3*n). A131961(n) = a(3*n). A112608(n) = a(3*n + 1).

Example

1 + x + x^3 + 3*x^6 + 2*x^7 + 2*x^9 + x^10 + 2*x^12 + x^15 + 2*x^16 + ...
q + q^9 + q^25 + 3*q^49 + 2*q^57 + 2*q^73 + q^81 + 2*q^97 + q^121 + 2*q^129 + ...
a(6) = 3 since we can write 6 = 6*1^2 + 0 = 6*(-1)^2 + 0 = 0 + 6.

Mathematica

a[ n_] := If[ n < 0, 0, Sum[ KroneckerSymbol[ -3, d], {d, Divisors[ 8 n + 1]}]] (* Michael Somos, Jun 16 2011 since V6 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ EllipticTheta[ 3, 0, q^6] EllipticTheta[ 2, 0, q^(1/2)] / (2 q^(1/8)), {q, 0, n}]] (* Michael Somos, Jun 16 2011 *)

Prog

(PARI) {a(n) = if( n<0, 0, n = 8*n + 1; sumdiv(n, d, kronecker(-3, 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)^2 * eta(x^12 + A)^5 / (eta(x + A) * eta(x^6 + A)^2 * eta(x^24 + A)^2), n))} /* Michael Somos, Sep 29 2006 */

Crossrefs

Cf. A112604, A112608, A123484, A131961.

Sequence in context: A145878 A195662 A340860 * A108512 A054503 A281451

Adjacent sequences: A112603 A112604 A112605 * A112607 A112608 A112609

Keyword

nonn

Author

James A. Sellers, Dec 21 2005

Status

approved