A214293 a(n) = 1 if n is a square, -1 if n is five times a square.

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

Offset

1,1

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Shaun Cooper and Michael Hirschhorn, On some infinite product identities, Rocky Mountain J. Math., 31 (2001) 131-139. see p. 134 Theorem 4.

Michael Somos, Introduction to Ramanujan theta functions.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Formula

Expansion of (phi(q) - phi(q^5)) / 2 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Sep 24 2013

Expansion of q * f(q^3, q^7) * f(-q^4, -q^16) / f(-q^8, -q^12) in powers of q where f() is Ramanujan's two-variable theta function.

Expansion of q * f(x, x^9) * f(-q, -q^4) / f(-q^2, -q^3) in powers of q where f() is Ramanujan's two-variable theta function. - Michael Somos, Sep 24 2013

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

Multiplicative with a(5^e) = (-1)^e, a(p^e) = 1 if e even, 0 otherwise.

G.f.: (theta_3(q) - theta_3(q^5)) / 2 = 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) = a(n). a(5*n) = -a(n).

a(4*n) = A214293(n). a(4*n+1) = A214960(n). - Michael Somos, Sep 24 2013

Sum_{k=1..n} a(k) ~ c*sqrt(n), where c = 1 - 1/sqrt(5) = 0.5527864... (A322159). - Amiram Eldar, Oct 24 2023

Example

G.f. = q + q^4 - q^5 + q^9 + q^16 - q^20 + q^25 + q^36 - q^45 + q^49 + q^64 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] - EllipticTheta[ 3, 0, q^5]) / 2,  {q, 0, n}];
a[ n_] := If[ n < 0, 0, Boole[ OddQ [ Length @ Divisors @ n]] - Boole[ OddQ [ Length @ Divisors [5 n]]]];

Prog

(PARI) {a(n) = issquare(n) - issquare(5*n)};
(PARI) {a(n) = if( n<1, 0, direuler( p=2, n, if( p==5, 1 - X, 1) / (1 - X^2 ))[n])};
(Magma) Basis( ModularForms( Gamma1(20), 1/2), 65) [2]; /* Michael Somos, Jul 01 2014 */

Crossrefs

Cf. A214284, A214295, A214960, A244612, A245485, A322159.

Sequence in context: A306862 A363801 A347283 * A323096 A359379 A120527

Adjacent sequences: A214290 A214291 A214292 * A214294 A214295 A214296

Keyword

sign,mult,easy

Author

Michael Somos, Jul 10 2012

Status

approved