A020501 Cyclotomic polynomials at x=-2.

-2, -3, -1, 3, 5, 11, 7, 43, 17, 57, 31, 683, 13, 2731, 127, 331, 257, 43691, 73, 174763, 205, 5419, 2047, 2796203, 241, 1016801, 8191, 261633, 3277, 178956971, 151, 715827883, 65537, 1397419, 131071, 24214051

Offset: 0

Author

Simon Plouffe

Keywords

• sign

Comments

a(0) depends on the definition of the 0th cyclotomic polynomial; Maple defines it as x, but Mathematica defines it as 1. - T. D. Noe, Jul 23 2008 [a(0) = x is correct. - N. J. A. Sloane, Aug 01 2008]

A020501[2n] = A019320[n] for all odd n > 1. (Because if m > 1 is odd, then Phi_2m(x) = Phi_m(-x) as demonstrated by Bloom). - Antti Karttunen, Aug 02 2001

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• D. M. Bloom, On the Coefficients of the Cyclotomic Polynomials, Amer.Math.Monthly 75, 372-377, 1968.
• Index entries for cyclotomic polynomials, values at X

Crossrefs

Cf. A020500, A020513.

Cf. A105603

Programs

Maple

with(numtheory,cyclotomic); f := n->subs(x=-2,cyclotomic(n,x)); seq(f(i),i=0..64);

Mathematica

Join[{-2}, Cyclotomic[Range[50], -2]] (* Paolo Xausa, Feb 26 2024 *)

PARI

a(n) = if (n, polcyclo(n, -2), -2); \\ Michel Marcus, Mar 05 2016