A244612 a(n) = 1 if n is a square, -1 if n is six times a square, 0 if n < 1.

1, 0, 0, 1, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0

Offset

1,1

Comments

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Michael Somos, Introduction to Ramanujan theta functions.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Formula

Expansion of (phi(q) - phi(q^6)) / 2 in powers of q where phi() is a Ramanujan theta function.

G.f.: (theta_3(q) - theta_3(q^6)) / 2 = Sum_{k>0} x^(k^2) - x^(6*k^2).

a(3*n) = A089801(n). a(3*n + 2) = 0.

Sum_{k=1..n} a(k) ~ c*sqrt(n), where c = 1 - 1/sqrt(6) = 0.5917517... . - Amiram Eldar, Oct 24 2023

Example

G.f. = q + q^4 - q^6 + q^9 + q^16 - q^24 + q^25 + q^36 + q^49 - q^54 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] - EllipticTheta[ 3, 0, q^6]) / 2,  {q, 0, n}];
a[ n_] := If[ n < 0, 0, Boole[ OddQ [ Length @ Divisors @ n]] - Boole[ OddQ [ Length @ Divisors [6 n]]]];

Prog

(PARI) {a(n) = issquare(n) - issquare(6*n)};
(Magma) Basis( ModularForms( Gamma1(24), 1/2), 64) [2];

Crossrefs

Cf. A089801, A214293, A214295, A245485.

Sequence in context: A352569 A101266 A260552 * A262805 A213728 A105367

Adjacent sequences: A244609 A244610 A244611 * A244613 A244614 A244615

Keyword

sign

Author

Michael Somos, Jul 01 2014

Extensions

More terms from Antti Karttunen, Dec 15 2017

Status

approved