A108563 Number of representations of n as sum of twice a square plus thrice a square.

1, 0, 2, 2, 0, 4, 0, 0, 2, 0, 0, 4, 2, 0, 4, 0, 0, 0, 2, 0, 4, 4, 0, 0, 0, 0, 0, 2, 0, 4, 4, 0, 2, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 2, 0, 6, 0, 0, 4, 0, 0, 4, 0, 0, 4, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 0, 0, 2, 0, 0, 6, 0, 8, 0, 0, 4, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 6, 4, 0, 4

Offset

0,3

Comments

Number of solutions to n = 2*a^2 + 3*b^2 in integers.

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

a(n) > 0 if and only if n is in A002480. a(n) < 2 if n is in A002481. - Michael Somos, Mar 01 2011

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

A115660(n) = A000377(n) - a(n). - 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

Cf. A000377, A002480, A002481, A115660.

Sequence in context: A342987 A090657 A167001 * A138476 A131381 A295215

Adjacent sequences: A108560 A108561 A108562 * A108564 A108565 A108566

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