The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A028586 Theta series of lattice with Gram matrix [2 1; 1 3]. 4
 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 (list; graph; refs; listen; history; text; internal format)
 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: A138758 A107501 A126732 * A253179 A300723 A263788 Adjacent sequences:  A028583 A028584 A028585 * A028587 A028588 A028589 KEYWORD nonn AUTHOR STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified July 26 17:37 EDT 2021. Contains 346294 sequences. (Running on oeis4.)