A230003 Array of coefficients of numerator polynomials of the rational function p(n, x + 1/x), where p(n,x) is the n-th cyclotomic polynomial.

1, 1, -1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 0, 3, 0, 1, 1, 1, 5, 4, 9, 4, 5, 1, 1, 1, -1, 3, -1, 1, 1, 1, 7, 6, 20, 14, 29, 14, 20, 6, 7, 1, 1, 1, 0, 4, 0, 7, 0, 4, 0, 1, 1, 0, 6, 1, 15, 3, 21, 3, 15, 1, 6, 0, 1, 1, -1, 5, -4, 9, -4, 5, -1, 1, 1, 1, 11, 10, 54

Offset

0,10

Comments

Define q(n,x) = p(n, x + 1/x). If r is a zero of p(n,x) then (1/2)*(r +- sqrt(r^2 - 4)) are zeros of q(n,x).

Links

Table of n, a(n) for n=0..79.

Example

First 6 rows:
  1
  1, -1,  1
  1,  1,  1
  1,  1,  3,  1,  1
  1,  0,  3,  0,  1
  1,  1,  5,  4,  9,  4,  5,  1,  1
First 4 polynomials:
  1,
  1 - x + x^2,
  1 + x + x^2,
  1 + x + 3*x^2 + x^3 + x^4.

Mathematica

z = 60; p[n_, x_] := p[x] = Cyclotomic[n, x]; Table[p[n, x], {n, 0, z/4}]; f1[n_, x_] := f1[n, x] = Numerator[Factor[p[n, x] /. x -> x + 1/x]]; Table[Expand[f1[n, x]], {n, 0, z/4}]
t = Flatten[Table[CoefficientList[f1[n, x], x], {n, 0, z/4}]]

Crossrefs

Cf. A231146.

Sequence in context: A241499 A236774 A110245 * A136093 A206831 A304222

Adjacent sequences: A230000 A230001 A230002 * A230004 A230005 A230006

Keyword

tabf,sign,easy

Author

Clark Kimberling, Nov 07 2013

Status

approved