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A010891 Inverse of 5th cyclotomic polynomial.

%I A010891 #43 Apr 03 2025 23:05:15
%S A010891 1,-1,0,0,0,1,-1,0,0,0,1,-1,0,0,0,1,-1,0,0,0,1,-1,0,0,0,1,-1,0,0,0,1,
%T A010891 -1,0,0,0,1,-1,0,0,0,1,-1,0,0,0,1,-1,0,0,0,1,-1,0,0,0,1,-1,0,0,0,1,-1,
%U A010891 0,0,0,1,-1,0,0,0,1,-1,0,0,0,1,-1,0,0,0,1
%N A010891 Inverse of 5th cyclotomic polynomial.
%C A010891 D(n):= a(n+3) appears in the formula 2*exp(2*Pi*n*i/5) = (A(n) + B(n)*phi) + (C(n) + D(n)*phi)*sqrt(2 + phi)*i, with the golden section phi, i = sqrt(-1) and A(n) = A164116(n+5), B(n) = A080891(n) and C(n) = A156174(n+4) for n >= 0. See one of the comments on A164116. - _Wolfdieter Lang_, Feb 26 2014
%C A010891 Periodic with period length 5. - _Ray Chandler_, Apr 03 2017
%H A010891 Vincenzo Librandi, <a href="/A010891/b010891.txt">Table of n, a(n) for n = 0..1000</a>
%H A010891 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-1,-1,-1,-1).
%H A010891 <a href="/index/Pol#poly_cyclo_inv">Index to sequences related to inverse of cyclotomic polynomials</a>
%F A010891 G.f.: 1/ ( 1+x+x^2+x^3+x^4 ). - _R. J. Mathar_, Mar 11 2011
%p A010891 with(numtheory,cyclotomic); c := n->series(1/cyclotomic(n,x),x,80);
%p A010891 # alternative
%p A010891 A010891 := proc(n)
%p A010891     op(1+(n mod 5),[1,-1,0,0,0]) ;
%p A010891 end proc:
%p A010891 seq(A010891(n),n=0..20) ; # _R. J. Mathar_, Feb 27 2025
%t A010891 CoefficientList[Series[1/Cyclotomic[5, x], {x, 0, 100}], x] (* _Vincenzo Librandi_, Apr 03 2014 *)
%o A010891 (PARI) Vec(1/polcyclo(5)+O(x^99)) \\ _Charles R Greathouse IV_, Mar 24 2014
%o A010891 (Magma) &cat[[1,-1,0,0,0]: n in [0..20]]; // _Vincenzo Librandi_, Apr 03 2014
%K A010891 sign,easy
%O A010891 0,1
%A A010891 _Simon Plouffe_