A246928 Number of integer solutions to x^2 + 3*y^2 + 3*z^2 = n.

1, 2, 0, 4, 10, 0, 4, 16, 0, 2, 8, 0, 12, 8, 0, 16, 26, 0, 0, 24, 0, 8, 8, 0, 20, 10, 0, 4, 32, 0, 8, 48, 0, 8, 16, 0, 10, 8, 0, 32, 40, 0, 8, 24, 0, 0, 16, 0, 28, 18, 0, 24, 40, 0, 4, 64, 0, 8, 8, 0, 32, 24, 0, 16, 58, 0, 16, 24, 0, 16, 16, 0, 0, 16, 0, 28

Offset

0,2

Comments

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

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

Formula

Expansion of phi(q) * phi(q^3)^2 in powers of q where phi() is a Ramanujan theta function.

Expansion of eta(q^2)^5 * eta(q^6)^10 / (eta(q)^2 * eta(q^3)^4 * eta(q^4)^2 * eta(q^12)^4) in powers of q.

Euler transform of period 12 sequence [ 2, -3, 6, -1, 2, -9, 2, -1, 6, -3, 2, -3, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 24^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A034933.

G.f. theta_3(q) * theta_3(q^3)^2.

a(3*n) = A034933(n). a(3*n + 1) = 2 * A246926(n). a(3*n + 2) = 0.

Example

G.f. = 1 + 2*q + 4*q^3 + 10*q^4 + 4*q^6 + 16*q^7 + 2*q^9 + 8*q^10 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^3]^2, {q, 0, n}]; (* Michael Somos, Jan 08 2015 *)

Prog

(PARI) {a(n) = if( n<1, n==0, qfrep( [ 1, 0, 0; 0, 3, 0; 0, 0, 3], n)[n] * 2)};
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^6 + A)^10 / (eta(x + A)^2 * eta(x^3 + A)^4 * eta(x^4 + A)^2 * eta(x^12 + A)^4), n))};
(Magma) A := Basis( ModularForms( Gamma1(12), 3/2), 76); A[1] + 2*A[2] + 4*A[4] + 10*A[5];

Crossrefs

Cf. A034933, A246926.

Sequence in context: A002938 A111938 A224822 * A167341 A359188 A361269

Adjacent sequences: A246925 A246926 A246927 * A246929 A246930 A246931

Keyword

nonn

Author

Michael Somos, Sep 07 2014

Status

approved