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
KEYWORD
nonn
AUTHOR
STATUS
approved