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Inverse of 8th cyclotomic polynomial.
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%I #50 Dec 14 2023 05:51:07

%S 1,0,0,0,-1,0,0,0,1,0,0,0,-1,0,0,0,1,0,0,0,-1,0,0,0,1,0,0,0,-1,0,0,0,

%T 1,0,0,0,-1,0,0,0,1,0,0,0,-1,0,0,0,1,0,0,0,-1,0,0,0,1,0,0,0,-1,0,0,0,

%U 1,0,0,0,-1,0,0,0,1,0,0,0,-1,0,0,0,1

%N Inverse of 8th cyclotomic polynomial.

%C Periodic with period length 8. - _Ray Chandler_, Apr 03 2017

%H Vincenzo Librandi, <a href="/A014017/b014017.txt">Table of n, a(n) for n = 0..1000</a>

%H John M. Campbell, <a href="http://arxiv.org/abs/1105.3399">An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences</a>, arXiv preprint arXiv:1105.3399 [math.GM], 2011.

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

%H <a href="/index/Pol#poly_cyclo_inv">Index to sequences related to inverse of cyclotomic polynomials</a>

%F a(4n) = (-1)^n, else a(n) = 0.

%F G.f.: 1/ ( 1+x^4 ). - _R. J. Mathar_, Mar 11 2011

%F a(n) = sin((sin(Pi*(n+1)/2)^2)*Pi*(n+2)/4). - _Mikael Aaltonen_, Jan 02 2015

%F E.g.f.: cos(x/sqrt(2))*cosh(x/sqrt(2)). - _Vaclav Kotesovec_, Feb 15 2015

%p with(numtheory,cyclotomic); c := n->series(1/cyclotomic(n,x),x,80);

%t CoefficientList[Series[1/Cyclotomic[8, x], {x, 0, 100}], x] (* _Vincenzo Librandi_, Apr 03 2014 *)

%o (PARI) Vec(1/polcyclo(8)+O(x^99)) \\ _Charles R Greathouse IV_, Mar 24 2014

%o (Magma) &cat[[1,0,0,0,-1,0,0,0]: n in [0..20]]; // _Vincenzo Librandi_, Apr 03 2014

%K sign,easy

%O 0,1

%A _Simon Plouffe_