A329290 Number of ordered triples (i, j, k) of integers such that n = i^2 + 4*j^2 + 4*k^2.

1, 2, 0, 0, 6, 8, 0, 0, 12, 10, 0, 0, 8, 8, 0, 0, 6, 16, 0, 0, 24, 16, 0, 0, 24, 10, 0, 0, 0, 24, 0, 0, 12, 16, 0, 0, 30, 8, 0, 0, 24, 32, 0, 0, 24, 24, 0, 0, 8, 18, 0, 0, 24, 24, 0, 0, 48, 16, 0, 0, 0, 24, 0, 0, 6, 32, 0, 0, 48, 32, 0, 0, 36, 16, 0, 0, 24, 32

Offset: 0

Author

Michael Somos, Nov 17 2019

Keywords

• nonn

Comments

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2^(5/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A169783.

Examples

			G.f. = 1 + 2*x + 6*x^4 + 8*x^5 + 12*x^8 + 10*x^9 + 8*x^12 + 8*x^13 + ...
		

Links

• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A005875, A045834, A169783.

Programs

Magma

A := Basis( ModularForms( Gamma0(16), 3/2), 77); A[1] + 2*A[2];

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[3, 0, x] EllipticTheta[3, 0, x^4]^2, {x, 0, n}];

PARI

{a(n) = my(A); if( n < 0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^8 + A)^10 / (eta(x + A)^2 * eta(x^4 + A)^6 * eta(x^16 + A)^4), n))};

Formula

Euler transform of period 16 sequence [2, -3, 2, 3, 2, -3, 2, -7, 2, -3, 2, 3, 2, -3, 2, -3, ...].

Expansion of phi(x) * phi(x^4)^2 = phi(x^4)^3 + 2*x*phi(x^4)*psi(x^4)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

G.f.: theta_3(q) * theta_3(q^4)^2, where theta_3() is the Jacobi theta function.

a(4*n) = A005875(n). a(4*n + 1) = 2 * A045834(n). a(4*n + 2) = a(4*n + 3) = 0.