A010891 Inverse of 5th cyclotomic polynomial.

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

Offset: 0

Author

Simon Plouffe

Keywords

• sign
• easy

Comments

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

Periodic with period length 5. - Ray Chandler, Apr 03 2017

Links

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

Programs

Magma

&cat[[1,-1,0,0,0]: n in [0..20]]; // Vincenzo Librandi, Apr 03 2014

Maple

with(numtheory,cyclotomic); c := n->series(1/cyclotomic(n,x),x,80);
# alternative
A010891 := proc(n)
    op(1+(n mod 5),[1,-1,0,0,0]) ;
end proc:
seq(A010891(n),n=0..20) ; # R. J. Mathar, Feb 27 2025

Mathematica

CoefficientList[Series[1/Cyclotomic[5, x], {x, 0, 100}], x] (* Vincenzo Librandi, Apr 03 2014 *)

PARI

Vec(1/polcyclo(5)+O(x^99)) \\ Charles R Greathouse IV, Mar 24 2014

Formula

G.f.: 1/ ( 1+x+x^2+x^3+x^4 ). - R. J. Mathar, Mar 11 2011