A033683 a(n) = 1 if n is an odd square not divisible by 3, otherwise 0.

0, 1, 0, 0, 0, 0, 0, 0, 0, 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, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset

0,1

References

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 105, Eq. (41).

Links

Antti Karttunen, Table of n, a(n) for n = 0..65537

Index entries for characteristic functions.

Formula

Essentially the series psi_6(z)=(1/2)(theta_2(z/9)-theta_2(z)).

a(A104777(n)) = 1.

A080995(n) = a(24n+1).

Multiplicative with a(p^e) = 1 if 2 divides e and p > 3, 0 otherwise. - Mitch Harris, Jun 09 2005

Euler transform of a period 144 sequence. - Michael Somos, Jan 26 2008

a(n) = A033684(n) * A000035(n).

Dirichlet g.f.: zeta(2*s) *(1-2^(-2s)) *(1-3^(-2s)). - R. J. Mathar, Mar 10 2011

G.f.: Sum_{k in Z} x^(6*k+1)^2. - Michael Somos, Dec 07 2019

Sum_{k=1..n} a(k) ~ sqrt(n)/3. - Amiram Eldar, Jan 14 2024

Example

G.f. = x + x^25 + x^49 + x^121 + x^169 + x^289 + x^361 + x^529 + x^625 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, x^4] - EllipticTheta[ 2, 0, x^36])/2, {x, 0, n}] // PowerExpand; (* Michael Somos, Dec 07 2019 *)
Table[If[OddQ[n]&&IntegerQ[Sqrt[n]]&&Mod[n, 3]!=0, 1, 0], {n, 0, 120}] (* Harvey P. Dale, Sep 06 2020 *)

Prog

(PARI) {a(n) = if( n%24 == 1, issquare(n), 0)}; /* Michael Somos, Jan 26 2008 */
(Haskell)
a033683 n = fromEnum $ odd n && mod n 3 > 0 && a010052 n == 1
-- Reinhard Zumkeller, Nov 14 2015
(Magma) Basis( ModularForms( Gamma0(144), 1/2), 106)[2]; /* Michael Somos, Dec 07 2019 */

Crossrefs

Cf. A098108, A033684.

Cf. A010052, A010872, A104777.

Sequence in context: A178333 A063524 A326168 * A216284 A360109 A355452

Adjacent sequences: A033680 A033681 A033682 * A033684 A033685 A033686

Keyword

nonn,mult,easy

Author

N. J. A. Sloane

Status

approved