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

Author

Paul Barry, Jul 14 2005

Keywords

• easy
• sign
• mult

Comments

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

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).

Crossrefs

Cf. A002605, A010892, A014021, A102587, A322796.

Programs

Magma

A110161:= func< n | KroneckerSymbol(12, n) >;
[A110161(n): n in [0..120]]; // G. C. Greubel, Oct 23 2024

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 *)

PARI

{a(n) = kronecker( 12, n)}; /* Michael Somos, Jun 11 2007 */

SageMath

def A110161(n): return kronecker(12, n)
[A110161(n) for n in range(121)] # G. C. Greubel, Oct 23 2024

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

Extensions

Corrected by T. D. Noe, Dec 12 2006