|
%I #54 Dec 18 2023 10:16:30
%S 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 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 0,0,0,-1,-1,0,0,0,1,1,0,0,0,-1,-1,0,0,0,1
%N Inverse of 10th cyclotomic polynomial.
%C The g.f. is the image of the g.f. of Fib(n+1) by the transform A(x) -> (1/(1 + x^2)^2)*A(x/(1+x^2)). The denominator is associated with the knots 4_1 and 5_1 by their Alexander and Jones polynomials respectively. - _Paul Barry_, Oct 16 2004
%C For n >= 1, a(n) is the determinant of an n X n Toeplitz matrix M satisfying: M(i,j) = 1 if -1 <= j - i <= 3 and 0 otherwise. - _Dmitry Efimov_, Jun 23 2015
%C Period 10: repeat [1,1,0,0,0,-1,-1,0,0,0]. - _Wesley Ivan Hurt_, Jun 24 2015
%C This is the Riordan transform of {A000045(n+1)}, n >= 0, with the Riordan matrix A049310 (Chebyshev S) of the Bell type. See the first comment by _Paul Barry_. - _Wolfdieter Lang_, Feb 18 2017
%H Vincenzo Librandi, <a href="/A014019/b014019.txt">Table of n, a(n) for n = 0..1000</a>
%H Dror Bar-Natan, <a href="http://katlas.org/wiki/Main_Page">The Rolfsen Knot Table</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,-1,1,-1).
%H <a href="/index/Pol#poly_cyclo_inv">Index to sequences related to inverse of cyclotomic polynomials</a>
%F G.f.: 1/(1 - x + x^2 - x^3 + x^4). - _Paul Barry_, Oct 16 2004
%F a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4), n > 4. - _Wesley Ivan Hurt_, Jun 24 2015
%p with(numtheory,cyclotomic); c := n->series(1/cyclotomic(n,x),x,80);
%t CoefficientList[Series[1/Cyclotomic[10, x], {x, 0, 100}], x] (* _Vincenzo Librandi_, Apr 03 2014 *)
%o (PARI) Vec(1/polcyclo(10)+O(x^99)) \\ _Charles R Greathouse IV_, Mar 24 2014
%o (Magma) &cat[[1,1,0,0,0,-1,-1,0,0,0]: n in [0..15]]; // _Vincenzo Librandi_, Apr 03 2014
%Y Cf. A000045, A049310, A099443, A281861.
%K sign,easy
%O 0,1
%A _Simon Plouffe_
|
