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

%I A110161 #34 Oct 25 2024 06:32:06
%S A110161 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 A110161 -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 A110161 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 A110161 Expansion of x*(1-x^2)/(1-x^2+x^4).
%C A110161 Transform of A002605 by the Riordan array A102587. Denominator is the 12th cyclotomic polynomial.
%H A110161 Antti Karttunen, <a href="/A110161/b110161.txt">Table of n, a(n) for n = 0..65537</a>
%H A110161 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,-1).
%F A110161 Periodic of length 12: 0, 1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 1. - _T. D. Noe_, Dec 12 2006
%F A110161 From _Michael Somos_, Jun 11 2007: (Start)
%F A110161 Euler transform of length 12 sequence [0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 1].
%F A110161 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).
%F A110161 a(n) = a(-n) = -a(n + 6) for all n in Z.
%F A110161 G.f.: x * (1 - x^4) * (1 - x^6) / (1 - x^12). (End)
%F A110161 a(2*n - 1) = A010892(n). - _Michael Somos_, Jan 29 2015
%F A110161 a(n) = A014021(n+1). - _R. J. Mathar_, Nov 13 2023
%t A110161 a[ n_] := JacobiSymbol[ 12, n]; (* _Michael Somos_, Jan 29 2015 *)
%t A110161 LinearRecurrence[{0,1,0,-1},{0,1,0,0},110] (* _Harvey P. Dale_, Jul 11 2015 *)
%o A110161 (PARI) {a(n) = kronecker( 12, n)}; /* _Michael Somos_, Jun 11 2007 */
%o A110161 (Magma)
%o A110161 A110161:= func< n | KroneckerSymbol(12, n) >;
%o A110161 [A110161(n): n in [0..120]]; // _G. C. Greubel_, Oct 23 2024
%o A110161 (SageMath)
%o A110161 def A110161(n): return kronecker(12, n)
%o A110161 [A110161(n) for n in range(121)] # _G. C. Greubel_, Oct 23 2024
%Y A110161 Cf. A002605, A010892, A014021, A102587, A322796.
%K A110161 easy,sign,mult
%O A110161 0,1
%A A110161 _Paul Barry_, Jul 14 2005
%E A110161 Corrected by _T. D. Noe_, Dec 12 2006