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

0,3

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

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

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

Example

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 + ...

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 *)

Prog

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

Crossrefs

Sequence in context: A126732 A371648 A347679 * A253179 A300723 A263788

Adjacent sequences: A028583 A028584 A028585 * A028587 A028588 A028589

Keyword

nonn

Author

N. J. A. Sloane

Status

approved