A028586 Theta series of lattice with Gram matrix [2 1; 1 3].

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Ramanujan theta functions: f(q) := Prod_{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..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

The number of integer solutions (x, y) to 2*x^2 + 2*x*y + 3*y^2 = n, discriminant -20. - Ray Chandler, Jul 12 2014

Examples

			1 + 2*q^2 + 4*q^3 + 4*q^7 + 2*q^8 + 2*q^10 + 4*q^12 + 4*q^15 + 6*q^18 + 4*q^23 + 8*q^27 + 4*q^28 + 2*q^32 + 4*q^35 + 2*q^40 + 8*q^42 + 4*q^43 + 4*q^47 + ...
		

Links

• John Cannon, 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, page 8 equation (3.18).
• 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

Programs

Mathematica

terms = 104; phi[q_] := EllipticTheta[3, 0, q]; chi[q_] := ((1 - InverseEllipticNomeQ[q])*InverseEllipticNomeQ[q]/(16*q))^(-1/24); psi[q_] := (1/2)*q^(-1/8)*EllipticTheta[2, 0, q^(1/2)]; s = phi[q^2]*phi[q^10] + 4*q^3*psi[q^4]*psi[q^20] + O[q]^(terms+1); CoefficientList[s, q] (* Jean-François Alcover, Jul 04 2017, after Michael Somos *)
r[n_]:=Reduce[{x, y}.{{2, 1}, {1, 3}}.{x, y}==n, {x, y}, Integers]; Table[rn=r[n]; Which[rn===False, 0, Head[rn]===Or, Length[rn], Head[rn]===And, 1], {n, 0, 105}] (* Vincenzo Librandi, Feb 23 2020 *)

PARI

{a(n) = if( n<1, n==0, qfrep([2, 1; 1, 3], n)[n] * 2)} /* Michael Somos, Aug 13 2006 */

Formula

G.f.: Sum_{n,m} x^(2*n^2 + 2*m*n + 3*m^2). - Michael Somos, Jan 31 2011

Expansion of (theta_3(z)*theta_3(5z)+theta_2(z)*theta_2(5z)).

Expansion of phi(q^2) * phi(q^10) + 4 * q^3 * psi(q^4) * psi(q^20) in powers of q where phi(q),psi(q) are Ramanujan theta functions. - Michael Somos, Aug 13 2006

If p is prime then a(p) is nonzero iff p is in A106865.

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