A033718 Product theta3(q^d); d | 5.

1, 2, 0, 0, 2, 2, 4, 0, 0, 6, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 2, 8, 0, 0, 4, 2, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 4, 0, 0, 0, 6, 4, 0, 0, 6, 0, 0, 0, 0, 8, 0, 4, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 0, 0, 8, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 10, 0, 0, 8, 0, 4, 0, 0, 4, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 2, 4, 0, 0, 0, 8, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 4, 2, 0, 0, 0, 2, 12

Offset

0,2

Comments

Ramanujan theta functions: f(q) := Product_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k>=0} q^(k*(k+1)/2) (A010054), chi(q) := Product_{k>=0} (1+q^(2k+1)) (A000700).

Number of representations of n as a sum of five times a square and a square. - Ralf Stephan, May 14 2007

References

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p 102 eq 9.

Links

Robert Israel, Table of n, a(n) for n = 0..10000

A. Berkovich and H. Yesilyurt, Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms, arXiv:math/0611300 [math.NT], 2006-2007.

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Theta series of lattice with Gram matrix [1 0 / 0 5].

Expansion of phi(q)phi(q^5) in powers of q where phi(q) is a Ramanujan theta function.

Euler transform of period 20 sequence [ 2, -3, 2, -1, 4, -3, 2, -1, 2, -6, 2, -1, 2, -3, 4, -1, 2, -3, 2, -2, ...]. - Michael Somos, Aug 13 2006

If p is prime then a(p) is nonzero iff p is in A033205.

0=a(n)a(2n) and 2*A035170(n) = a(n) + a(2n) if n>0. - Michael Somos, Oct 21 2006

a(n) is nonzero iff n is in A020669. - Robert Israel, Dec 22 2015

Maple

S:= series(JacobiTheta3(0, q)*JacobiTheta3(0, q^5), q, 1001):
seq(coeff(S, q, j), j=0..1000); # Robert Israel, Dec 22 2015

Mathematica

terms = 127; s = EllipticTheta[3, 0, q] EllipticTheta[3, 0, q^5] + O[q]^terms; CoefficientList[s, q] (* Jean-François Alcover, Jul 04 2017 *)

Prog

(PARI) {a(n)=if(n<1, n==0, qfrep([1, 0; 0, 5], n)[n]*2)} /* Michael Somos, Aug 13 2006 */
(PARI)
N=666;  x='x+O('x^N);
T3(x)=1+2*sum(n=1, ceil(sqrt(N)), x^(n*n));
Vec(T3(x)*T3(x^5))
/* Joerg Arndt, Sep 21 2012 */

Crossrefs

Cf. A000700, A000122, A010054, A020669, A033205, A121373.

Sequence in context: A262904 A144377 A138527 * A033737 A033747 A087611

Adjacent sequences: A033715 A033716 A033717 * A033719 A033720 A033721

Keyword

nonn

Author

N. J. A. Sloane

Status

approved