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A004531 Number of integer solutions to x^2 + 4 * y^2 = n. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

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)

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

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).

EXAMPLE

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 + ...

MATHEMATICA

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

PROG

(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

CROSSREFS

Cf. A004018, A004020.

Sequence in context: A185146 A080964 A134014 * A072071 A045836 A182056

Adjacent sequences:  A004528 A004529 A004530 * A004532 A004533 A004534

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified November 19 00:12 EST 2018. Contains 317332 sequences. (Running on oeis4.)