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 A033718 Product theta3(q^d); d | 5. 5
 1, 2, 0, 0, 2, 2, 4, 0, 0, 6, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 2, 8, 0, 0, 4, 2, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 4, 0, 0, 0, 6, 4, 0, 0, 6, 0, 0, 0, 0, 8, 0, 4, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 0, 0, 8, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 10, 0, 0, 8, 0, 4, 0, 0, 4, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 2, 4, 0, 0, 0, 8, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 4, 2, 0, 0, 0, 2, 12 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Ramanujan theta functions: f(q) := Product_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k>=0} q^(k*(k+1)/2) (A010054), chi(q) := Product_{k>=0} (1+q^(2k+1)) (A000700). Number of representations of n as a sum of five times a square and a square. - Ralf Stephan, May 14 2007 REFERENCES J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p 102 eq 9. LINKS Robert Israel, Table of n, a(n) for n = 0..10000 A. Berkovich and H. Yesilyurt, Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms, arXiv:math/0611300 [math.NT], 2006-2007. N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references) Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Theta series of lattice with Gram matrix [1 0 / 0 5]. Expansion of phi(q)phi(q^5) in powers of q where phi(q) is a Ramanujan theta function. Euler transform of period 20 sequence [ 2, -3, 2, -1, 4, -3, 2, -1, 2, -6, 2, -1, 2, -3, 4, -1, 2, -3, 2, -2, ...]. - Michael Somos, Aug 13 2006 If p is prime then a(p) is nonzero iff p is in A033205. 0=a(n)a(2n) and 2*A035170(n) = a(n) + a(2n) if n>0. - Michael Somos, Oct 21 2006 a(n) is nonzero iff n is in A020669. - Robert Israel, Dec 22 2015 MAPLE S:= series(JacobiTheta3(0, q)*JacobiTheta3(0, q^5), q, 1001): seq(coeff(S, q, j), j=0..1000); # Robert Israel, Dec 22 2015 MATHEMATICA terms = 127; s = EllipticTheta[3, 0, q] EllipticTheta[3, 0, q^5] + O[q]^terms; CoefficientList[s, q] (* Jean-François Alcover, Jul 04 2017 *) PROG (PARI) {a(n)=if(n<1, n==0, qfrep([1, 0; 0, 5], n)[n]*2)} /* Michael Somos, Aug 13 2006 */ (PARI) N=666;  x='x+O('x^N); T3(x)=1+2*sum(n=1, ceil(sqrt(N)), x^(n*n)); Vec(T3(x)*T3(x^5)) /* Joerg Arndt, Sep 21 2012 */ CROSSREFS Cf. A000700, A000122, A010054, A020669, A033205, A121373. Sequence in context: A262904 A144377 A138527 * A033737 A033747 A087611 Adjacent sequences:  A033715 A033716 A033717 * A033719 A033720 A033721 KEYWORD nonn AUTHOR STATUS approved

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Last modified April 1 04:15 EDT 2020. Contains 333155 sequences. (Running on oeis4.)