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 A004531 Number of integer solutions to x^2 + 4 * y^2 = n. 7
 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) Michael 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-François 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 A329264 A045836 Adjacent sequences:  A004528 A004529 A004530 * A004532 A004533 A004534 KEYWORD nonn AUTHOR STATUS approved

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Last modified April 10 06:54 EDT 2021. Contains 342843 sequences. (Running on oeis4.)