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A214284 Characteristic function of squares or five times squares. 3
1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 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, 0, 0, 0, 0, 0, 1, 1, 0, 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).

A195198 is a similar sequence except with three instead of five. - 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. 134 Theorem 4.

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of f(q, q^9) * f(-q^8, -q^12) / f(-q^4, -q^16) in powers of q where f(, ) is Ramanujan's general theta function.

Expansion of f(q^3, q^7) * f(-q^2, -q^3) / f(-q, -q^4) in powers of q where f(, ) is Ramanujan's general theta function.

Euler transform of period 20 sequence [1, -1, 0, 1, 0, 0, 0, -1, 1, -1, 1, -1, 0, 0, 0, 1, 0, -1, 1, -1, ...].

a(n) is multiplicative with a(0) = a(5^e) = 1, a(p^e) = 1 if e is even, 0 otherwise.

G.f.: (theta_3(q) + theta_3(q^5)) / 2 = 1 + (Sum_{k>0} x^(k^2) + x^(5*k^2)).

Dirichlet g.f.: zeta(2*s) * (1 + 5^-s).

a(4*n + 2) = a(4*n + 3) = 0. a(4*n + 1) = A127693(n). a(5*n) = a(n).

EXAMPLE

G.f. = 1 + x + x^4 + x^5 + x^9 + x^16 + x^20 + x^25 + x^36 + x^45 + x^49 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ Series[ (EllipticTheta[ 3, 0, q] + EllipticTheta[ 3, 0, q^5]) / 2, {q, 0, n}], {q, 0, n}];

a[ n_] := If[ n < 0, 0, Boole[ OddQ [ Length @ Divisors @ n] || OddQ [ Length @ Divisors[5 n]]]];

PROG

(PARI) {a(n) = issquare(n) || issquare(5*n)};

(PARI) {a(n) = if( n<1, n==0, direuler( p=2, n, if( p==5, 1 + X, 1) / (1 - X^2))[n])};

CROSSREFS

Cf. A127693, A195198.

Sequence in context: A181115 A284527 A151666 * A191747 A330323 A280933

Adjacent sequences:  A214281 A214282 A214283 * A214285 A214286 A214287

KEYWORD

nonn,mult,easy

AUTHOR

Michael Somos, Jul 09 2012

STATUS

approved

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Last modified December 5 16:22 EST 2020. Contains 338954 sequences. (Running on oeis4.)