A033717 Number of integer solutions to the equation x^2 + 2*y^2 + 4*z^2 = n.

1, 2, 2, 4, 4, 4, 8, 8, 6, 6, 8, 4, 8, 12, 0, 8, 12, 8, 10, 12, 8, 8, 24, 8, 8, 14, 8, 16, 16, 4, 0, 16, 6, 16, 16, 8, 12, 20, 24, 8, 24, 8, 16, 20, 8, 20, 0, 16, 24, 18, 10, 8, 24, 12, 32, 24, 0, 16, 24, 12, 16, 20, 0, 24, 12, 8, 16, 28, 16, 16, 48, 8, 30, 32, 8, 20, 24, 16, 0, 16, 24, 18

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + 2*q + 2*q^2 + 4*q^3 + 4*q^4 + 4*q^5 + 8*q^6 + 8*q^7 + 6*q^8 + ...
		

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p 102 eq 9.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A004015, A008443, A014455, A033763, A045826, A045828, A045831, A045834, A213022, A213622, A213624, A213625, A246811.

Programs

Magma

A := Basis( ModularForms( Gamma1(16), 3/2), 82); A[1] + 2*A[2] + 2*A[3] + 4*A[4] + 4*A[5] + 4*A[6] + 8*A[7] + 8*A[8] + 6*A[9] + 8*A[10] + 4*A[11]; /* Michael Somos, Sep 03 2014 */

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2] EllipticTheta[ 3, 0, q^4], {q, 0, n}]; (* Michael Somos, Sep 03 2014 *)

PARI

{a(n) = my(G); if( n<0, 0, G = [1, 0, 0; 0, 2, 0; 0, 0, 4]; polcoeff( 1 + 2 * x * Ser(qfrep( G, n)), n))}; /* Michael Somos, Sep 03 2014 */

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^4 + A) * eta(x^8 + A)^3 / (eta(x + A)^2 * eta(x^16 + A)^2), n))}; /* Michael Somos, Sep 03 2014 */

Formula

Expansion of phi(q) * phi(q^2) * phi(q^4) in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Sep 03 2014

Euler transform of period 16 sequence [2, -1, 2, -2, 2, -1, 2, -5, 2, -1, 2, -2, 2, -1, 2, -3, ...]. - Michael Somos, Sep 03 2014

G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 8 (t/i)^(3/2) f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 03 2014

a(2*n + 1) = 2 * A045828(n). a(4*n) = A014455(n). a(4*n + 1) = 2 * A213625(n). a(4*n + 2) = 2 * A246811(n). a(4*n + 3) = 4 * A213624(n). - Michael Somos, Sep 03 2014

a(8*n) = A005875(n). a(8*n + 1) = 2 * A213622(n). a(8*n + 2) = 2 * A045834(n). a(8*n + 7) = 8 * A033763(n). - Michael Somos, Sep 03 2014

a(16*n) = A004015(n). a(16*n + 2) = 2 * A213022(n). a(16*n + 6) = 8 *

A008443(n). a(16*n + 8) = 2 * A045826(n). a(16*n + 10) = 8 * A045831(n). a(16*n + 14) = 0. - Michael Somos, Sep 03 2014

G.f.: theta_3(q) * theta_3(q^2) * theta_3(q^4).