OFFSET
0,3
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
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.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
G.f.: 1 + Sum_{k>0} x^k * (1 + x^(4*k)) * (1 + x^(6*k)) / (1 + x^(12*k)) - Sum_{k>0} Kronecker( k, 3) * x^k * (1 - x^(2*k)) / (1 + x^(4*k)).
G.f.: Sum_{i, j in Z} x^(2*i^2 + 3*j^2). - Michael Somos, Mar 01 2011
Expansion of phi(q^2) * phi(q^3) in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Mar 01 2011
Euler transform of period 24 sequence [0, 2, 2, -3, 0, -1, 0, -1, 2, 2, 0, -4, 0, 2, 2, -1, 0, -1, 0, -3, 2, 2, 0, -2, ...]. - Michael Somos, Jan 20 2017
Expansion of eta(q^4)^5 * eta(q^6)^5 / (eta(q^2)^2 * eta(q^3)^2 * eta(q^8)^2 * eta(q^12)^2) in powers of q. - Michael Somos, Jan 20 2017
EXAMPLE
G.f. = 1 + 2*x^2 + 2*x^3 + 4*x^5 + 2*x^8 + 4*x^11 + 2*x^12 + 4*x^14 + 2*x^18 + ...
a(0) = 1 since 0 = 2*0^2 + 3*0^2, a(5) = 4 since 5 = 2*1^2 + 3*1^2 = 2*(-1)^2 + 3*1^2 = 2*1^2 + 3*(-1)^2 = 2*(-1^2) + 3*(-1)^2.
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^2] EllipticTheta[ 3, 0, q^3], {q, 0, n}]; (* Michael Somos, Apr 19 2015 *)
a[n_] := Module[{a, b, r}, r = Reduce[n == 2a^2 + 3b^2, {a, b}, Integers]; Which[r === False, 0, r[[0]] === And, 1, r[[0]] === Or, Length[r]]];
Table[a[n], {n, 0, 105}] (* Jean-François Alcover, Jan 09 2019 *)
PROG
(PARI) for(n=0, 120, print1(if(n<1, n==0, qfrep([2, 0; 0, 3], n)[n]*2), ", "))
(PARI) {a(n) = my(G); if( n<0, 0, G = [2, 0; 0, 3]; polcoeff( 1 + 2 * x * Ser( qfrep( G, n)), n))}; /* Michael Somos, Mar 01 2011 */
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^5 * eta(x^6 + A)^5 / (eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^8 + A)^2 * eta(x^12 + A)^2), n))}; /* Michael Somos, Jan 20 2017 */
(Sage)
Q = DiagonalQuadraticForm(ZZ, [3, 2])
Q.representation_number_list(102) # Peter Luschny, Jun 20 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Ralf Stephan, May 13 2007
EXTENSIONS
Edited by Charles R Greathouse IV, Oct 28 2009
Edited by N. J. A. Sloane, Mar 04 2011
STATUS
approved