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A110161
Expansion of x*(1-x^2)/(1-x^2+x^4).
13
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.
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
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
KEYWORD
easy,sign,mult,changed
AUTHOR
Paul Barry, Jul 14 2005
EXTENSIONS
Corrected by T. D. Noe, Dec 12 2006
STATUS
approved