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

Author

N. J. A. Sloane

Keywords

• nonn

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

Crossrefs

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

Programs

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 *)

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 */

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

a(0) = 1, a(n) = (1+(-1)^t)b(n) for n > 0, where t is the number of prime factors of n, counting multiplicity, which are == 2,3,7 (mod 20), and b() is multiplicative with b(p^e) = (e+1) for primes p == 1,3,7,9 (mod 20) and b(p^e) = (1+(-1)^e)/2 for primes p == 11,13,17,19 (mod 20). (This formula is Corollary 3.3 in the Berkovich-Yesilyurt paper) - Jeremy Lovejoy, Nov 12 2024