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A195198 Characteristic function of squares or three times squares. 4
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 (list; graph; refs; listen; history; text; internal format)
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.

M. 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).

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]]]];

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: A137161 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

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Last modified October 16 12:35 EDT 2018. Contains 316263 sequences. (Running on oeis4.)