A195198 Characteristic function of squares or three times squares.

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

Offset: 0

Author

Michael Somos, Sep 11 2011

Keywords

• nonn
• mult
• easy

Comments

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

A214284 is a similar sequence except with five instead of three. - Michael Somos, Oct 22 2017

Examples

			G.f. = 1 + q + q^3 + q^4 + q^9 + q^12 + q^16 + q^25 + q^27 + q^36 + q^48 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• S. Cooper and M. Hirschhorn, On some infinite product identities, Rocky Mountain J. Math., 31 (2001) 131-139. See p. 133 Theorem 3.
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A089801, A127692, A145377, A214284.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ Series[ (EllipticTheta[ 3, 0, q] + EllipticTheta[ 3, 0, q^3]) / 2, {q, 0, n}], {q, 0, n}];
a[ n_] := If[ n < 0, 0, Boole[ OddQ [ Length @ Divisors @ n] || OddQ [ Length @ Divisors[3 n]]]];
Table[If[AnyTrue[{Sqrt[n],Sqrt[3n]},IntegerQ],1,0],{n,0,110}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 22 2020 *)

PARI

{a(n) = issquare(n) || issquare(3*n)};

PARI

{a(n) = if( n<1, n==0, direuler( p=2, n, if( p==3, 1 + X, 1) / (1 - X^2))[n])};

Formula

Euler transform of period 12 sequence [1, -1, 1, 0, 0, -1, 0, 0, 1, -1, 1, -1, ...].

Expansion of psi(q^3) * f(-q^2, -q^10) / f(-q, -q^11) in powers of q where psi(), is a Ramanujan theta function and f(, ) is Ramanujan's general theta function.

Multiplicative with a(0) = a(3^e) = 1, a(p^e) = 1 if e even, 0 otherwise.

G.f.: (theta_3(q) + theta_3(q^3)) / 2 = 1 + (Sum_{k>0} x^(k^2) + x^(3*k^2)).

Dirichlet g.f.: zeta(2*s) * (1 + 3^-s).

a(n) = A145377(n) unless n=0. a(3*n + 2) = 0. a(2*n + 1) = A127692(n). a(3*n) = a(n). a(3*n + 1) = A089801(n).

Sum_{k=0..n} a(k) ~ (1+1/sqrt(3)) * sqrt(n). - Amiram Eldar, Sep 14 2023