A014025 Expansion of the inverse of the 16th cyclotomic polynomial.

1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0

Offset: 0

Author

Simon Plouffe

Keywords

• sign
• easy

Comments

Period 16: repeat [1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0].

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, -1).
• Index to sequences related to inverse of cyclotomic polynomials

Programs

Magma

&cat[[1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0]: n in [0..8]]; // Vincenzo Librandi, Apr 03 2014

Maple

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

Mathematica

CoefficientList[Series[1/Cyclotomic[16, x], {x, 0, 100}], x] (* Vincenzo Librandi, Apr 03 2014 *)
LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, -1},{1, 0, 0, 0, 0, 0, 0, 0},99] (* Ray Chandler, Sep 15 2015 *)

PARI

Vec(1/polcyclo(16)+O(x^99)) \\ Charles R Greathouse IV, Mar 24 2014

Formula

G.f.: 1/(1 + x^8). - Klaus Brockhaus, May 17 2011

a(n) = -a(n-8).

abs(a(n)) = floor(1/2*cos(n*Pi/4)+1/2). - Gary Detlefs, May 16 2011

a(n)=(floor((n+8)/8)-floor((n+7)/8))*(-1)^floor(n/8). |a(n)| is the characteristic function of numbers that are multiples of 8. |a(n)|=floor(n/8)-floor((n-1)/8). - Boris Putievskiy, May 08 2013