A110161 Expansion of x*(1-x^2)/(1-x^2+x^4).

0, 1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 1, 0, 1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 1, 0, 1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 1, 0, 1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 1, 0, 1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 1, 0, 1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 1, 0, 1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 1, 0, 1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 1, 0, 1, 0, 0, 0, -1, 0, -1, 0

Offset

0,1

Comments

Transform of A002605 by the Riordan array A102587. Denominator is the 12th cyclotomic polynomial.

The Dirichlet character associated with the real quadratic field Q(sqrt(3)) (discriminant 12). - Jianing Song, Dec 13 2025

Links

Antti Karttunen, Table of n, a(n) for n = 0..65537

Index entries for linear recurrences with constant coefficients, signature (0,1,0,-1).

Formula

Periodic of length 12: 0, 1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 1. - T. D. Noe, Dec 12 2006

From Michael Somos, Jun 11 2007: (Start)

Euler transform of length 12 sequence [0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 1].

a(n) is multiplicative with a(2^e) = a(3^e) = 0^e, a(p^e) = 1 if p == 1, 11 (mod 12), a(p^e) = (-1)^e if p == 5, 7 (mod 12).

a(n) = a(-n) = -a(n + 6) for all n in Z.

G.f.: x * (1 - x^4) * (1 - x^6) / (1 - x^12). (End)

a(2*n - 1) = A010892(n). - Michael Somos, Jan 29 2015

a(n) = A014021(n+1). - R. J. Mathar, Nov 13 2023

E.g.f.: 2*cos(x/2)*sinh(sqrt(3)*x/2)/sqrt(3). - Amiram Eldar, Nov 27 2025

Mathematica

a[ n_] := JacobiSymbol[ 12, n]; (* Michael Somos, Jan 29 2015 *)
LinearRecurrence[{0, 1, 0, -1}, {0, 1, 0, 0}, 110] (* Harvey P. Dale, Jul 11 2015 *)

Prog

(PARI) {a(n) = kronecker( 12, n)}; /* Michael Somos, Jun 11 2007 */
(Magma)
A110161:= func< n | KroneckerSymbol(12, n) >;
[A110161(n): n in [0..120]]; // G. C. Greubel, Oct 23 2024
(SageMath)
def A110161(n): return kronecker(12, n)
[A110161(n) for n in range(121)] # G. C. Greubel, Oct 23 2024

Crossrefs

Cf. A002605, A010892, A014021, A102587.

Moebius transform of A035194.

Cf. A038874 (primes not inert in Q(sqrt(3))), A097933 (primes decomposing), A003630 (primes remaining inert).

Kronecker symbols {(D/n)} for negative fundamental discriminants D = -3..-47, -67, -163: A102283, A101455, A175629, A188510, A011582, A316569, A011585, A289741, A011586, A109017, A011588, A390614, A388073, A388072, A011591, A011592, A011596, A011615.

Kronecker symbols {(D/n)} for positive fundamental discriminants D = 5..41: A080891, A091337, this sequence, A011583, A011584, A322829, A322796, A390615, A011587, A391502, A011589, A391503, A011590.

Sequence in context: A379275 A322796 A109017 * A134667 A354354 A117943

Adjacent sequences: A110158 A110159 A110160 * A110162 A110163 A110164

Keyword

easy,sign,mult

Author

Paul Barry, Jul 14 2005

Extensions

Corrected by T. D. Noe, Dec 12 2006

Status

approved