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A110161 Expansion of x(1-x^2)/(1-x^2+x^4). 7

%I

%S 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,

%T -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,

%U 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

%N Expansion of x(1-x^2)/(1-x^2+x^4).

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

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,-1).

%F Periodic of length 12: 0, 1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 1. - _T. D. Noe_, Dec 12 2006

%F a(n)=(1/12)*{[n mod 12]-[(n+1) mod 12]-[(n+4) mod 12]+[(n+5) mod 12]-[(n+6) mod 12]+[(n+7) mod 12]+[(n+10) mod 12]-[(n+11) mod 12]}, with n>=0. - _Paolo P. Lava_, Jun 01 2007

%F Euler transform of length 12 sequence [ 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 1]. - _Michael Somos_, Jun 11 2007

%F 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). - _Michael Somos_, Jun 11 2007

%F G.f.: x * (1 - x^4) * (1 - x^6) / (1 - x^12). a(n) = a(-n) = -a(n + 6) for all n in Z. - _Michael Somos_, Jun 11 2007

%F a(2*n - 1) = A010892(n). - _Michael Somos_, Jan 29 2015

%t a[ n_] := JacobiSymbol[ 12, n]; (* _Michael Somos_, Jan 29 2015 *)

%t LinearRecurrence[{0,1,0,-1},{0,1,0,0},110] (* _Harvey P. Dale_, Jul 11 2015 *)

%o (PARI) {a(n) = kronecker( 12, n)}; /* _Michael Somos_, Jun 11 2007 */

%Y Cf. A010892.

%K easy,sign,mult

%O 0,1

%A _Paul Barry_, Jul 14 2005

%E Corrected by _T. D. Noe_, Dec 12 2006

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Last modified August 26 01:30 EDT 2019. Contains 326324 sequences. (Running on oeis4.)