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

Author

N. J. A. Sloane

Keywords

• nonn
• mult
• easy

Examples

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

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.

Crossrefs

Cf. A098108, A033684.

Cf. A010052, A010872, A104777.

Programs

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 */

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 *)

PARI

{a(n) = if( n%24 == 1, issquare(n), 0)}; /* Michael Somos, Jan 26 2008 */

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