A004531 Number of integer solutions to x^2 + 4 * y^2 = n.

1, 2, 0, 0, 4, 4, 0, 0, 4, 2, 0, 0, 0, 4, 0, 0, 4, 4, 0, 0, 8, 0, 0, 0, 0, 6, 0, 0, 0, 4, 0, 0, 4, 0, 0, 0, 4, 4, 0, 0, 8, 4, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 8, 4, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 4, 8, 0, 0, 8, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 8, 2, 0, 0, 0, 8, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 12, 4, 0, 0, 8

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*x + 4*x^4 + 4*x^5 + 4*x^8 + 2*x^9 + 4*x^13 + 4*x^16 + 4*x^17 + 8*x^20 + ...
		

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 120.
• B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 373 Entry 32.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
• Michael Somos, Introduction to Ramanujan theta functions.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Crossrefs

Cf. A004018, A004020, A019669.

Programs

Mathematica

CoefficientList[EllipticTheta[3, 0, x]*EllipticTheta[3, 0, x^4] + O[x]^105, x] (* Jean-François Alcover, Nov 05 2015 *)

PARI

{a(n) = if( n<1, n==0, 2 * qfrep([ 1, 0; 0, 4], n)[n])}; /* Michael Somos, Jul 04 2005 */

PARI

{a(n) = local(A, e1, e2, e4); if( n<0, 0, A = x * O(x^n); e1 = eta(x^2 + A); e2 = eta(x^4 + A); e4 = eta(x^8 + A); polcoeff( (e2^12 + e1^8 * e4^4 + 4 * x * e1^4 * e4^8) / (2 * e1^4 * e2^2 * e4^4), n))};

Sage

Q = DiagonalQuadraticForm(ZZ, [4, 1])
Q.representation_number_list(105) # Peter Luschny, Jun 20 2014

Formula

Expansion of (eta(q^2) * eta(q^8))^5 / (eta(q)^2 * eta(q^4)^4 * eta(q^16)^2) in powers of q.

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

Expansion of (theta2^2(q^2) + theta3^2(q^2) + theta4^2(q^2)) / 2 in powers of q.

Euler transform of period 16 sequence [ 2, -3, 2, 1, 2, -3, 2, -4, 2, -3, 2, 1, 2, -3, 2, -2, ...]. - Michael Somos, Jun 20 2014

G.f.: Sum_{i,j} x^(i^2 + 4 * j^2).

a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A004018(n). a(4*n + 1) = A004020(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/2 (A019669). - Amiram Eldar, Oct 15 2022