

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
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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) 131139. 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



