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

0,1

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

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

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

Example

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

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 *)

Prog

(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])};

Crossrefs

Cf. A089801, A127692, A145377, A214284.

Sequence in context: A329677 A077050 A128432 * A039966 A089451 A145099

Adjacent sequences: A195195 A195196 A195197 * A195199 A195200 A195201

Keyword

nonn,mult,easy

Author

Michael Somos, Sep 11 2011

Status

approved