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
KEYWORD
nonn
AUTHOR
STATUS
approved