A214509 a(n) = 1 if n is an odd square or twice a nonzero even square, -1 if a nonzero even square or twice an odd square else 0.

1, -1, 0, -1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 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, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 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. 133 Theorem 2.

Michael Somos, Introduction to Ramanujan theta functions.

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

Formula

Expansion of (- phi(-q) + phi(-q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.

Expansion of x * f(-x^2, -x^2) * f(x^2, x^14) / f(x, x^3) in powers of x where f() is Ramanujan's two-variable theta function.

a(n) is multiplicative with a(2) = -1, a(2^e) = (-1)^(e+1) if e>1, a(p^e) = (1 + (-1)^e) / 2 if p>2.

Euler transform of period 32 sequence [ -1, 0, -1, -1, -1, -1, -1, 0, -1, -1, -1, 0, -1, 0, -1, -1, -1, 0, -1, 0, -1, -1, -1, 0, -1, -1, -1, -1, -1, 0, -1, -1, ...].

G.f.: (theta_4(q^2) - theta_4(q)) / 2 = (Sum_{k>0} (-1)^k * (x^(2*k^2) - x^(k^2))).

a(n) = -(-1)^(n * (n + 1)/2) * A143259(n).

Dirichlet g.f.: (1 - 1/2^(2*s-1)) * (1 - 1/2^s) * zeta(2*s). - Amiram Eldar, Sep 12 2023

Example

G.f. = q - q^2 - q^4 + q^8 + q^9 - q^16 - q^18 + q^25 + q^32 - q^36 + q^49 - q^50 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q^2] - EllipticTheta[ 4, 0, q]) / 2, {q, 0, n}];
a[ n_] := If[ n < 0, 0, (-1)^(n (n + 1)/2) ( Boole @ OddQ[ Length @ Divisors[ 2 n]] - Boole @ OddQ[ Length @ Divisors[ n]])];

Prog

(PARI) {a(n) = (-1)^(n * (n + 1)/2) * (issquare(2*n) - issquare(n))};

Crossrefs

Cf. A000122, A000700, A010054, A121373, A143259.

Sequence in context: A359349 A266459 A354199 * A053866 A143259 A359818

Adjacent sequences: A214506 A214507 A214508 * A214510 A214511 A214512

Keyword

sign,easy,mult

Author

Michael Somos, Jul 19 2012

Status

approved