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%I #31 Feb 24 2020 04:30:05
%S 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,
%T 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,
%U 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
%N Theta series of lattice with Gram matrix [2 1; 1 3].
%C 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).
%C 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
%H John Cannon, <a href="/A028586/b028586.txt">Table of n, a(n) for n = 0..10000</a>
%H A. Berkovich and H. Yesilyurt, <a href="https://arxiv.org/abs/math/0611300">Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms</a>, arXiv:math/0611300 [math.NT], 2006-2007, page 8 equation (3.18).
%H N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F G.f.: Sum_{n,m} x^(2*n^2 + 2*m*n + 3*m^2). - _Michael Somos_, Jan 31 2011
%F Expansion of (theta_3(z)*theta_3(5z)+theta_2(z)*theta_2(5z)).
%F 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
%F If p is prime then a(p) is nonzero iff p is in A106865.
%F 0=a(n)a(2n) and 2*A035170(n)=a(n)+a(2n) if n>0. - _Michael Somos_, Oct 21 2006
%e 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 + ...
%t 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_ *)
%t 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 *)
%o (PARI) {a(n) = if( n<1, n==0, qfrep([2, 1; 1, 3], n)[n] * 2)} /* _Michael Somos_, Aug 13 2006 */
%K nonn
%O 0,3
%A _N. J. A. Sloane_
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