A230002 Array of coefficients of numerator polynomials of the rational function p(n, x - 1/x), where p(n,x) is the Fibonacci polynomial defined by p(1,x) = 1, p(2,x) = x, p(n,x) = x*p(n-1,x) + p(n-2,x).

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

Offset

0

Comments

Row n has 2*n-1 terms. 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). Appears to be a signed version of A071028.

Links

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

Example

First 4 rows:
   1;
  -1,  0,  1;
   1,  0, -1,  0,  1;
  -1,  0,  1,  0, -1,  0,  1;
First 4 polynomials:
   1;
  -1 + x^2;
   1 - x^2 + x^4;
  -1 + x^2 - x^4 + x^6;

Mathematica

p[n_, x_] := p[x] = Fibonacci[n, x]; Table[p[n, x], {n, 1, 10}]
f[n_, x_] := f[n, x] = Expand[Numerator[Factor[p[n, x] /. x -> x + 1/x]]]
g[n_, x_] := g[n, x] = Expand[Numerator[Factor[p[n, x] /. x -> x - 1/x]]]
h[n_, x_] := h[n, x] = Expand[Numerator[Factor[p[n, x] /. x -> x + 1 + 1/x]]]
t1 = Flatten[Table[CoefficientList[f[n, x], x], {n, 1, 12}]];  (* A229995 *)
t2 = Flatten[Table[CoefficientList[g[n, x], x], {n, 1, 12}]];  (* A230002 *)
t3 = Flatten[Table[CoefficientList[h[n, x], x], {n, 1, 12}]];  (* A059317 *)

Crossrefs

Cf. A229995.

Sequence in context: A338354 A014240 A014471 * A071028 A286987 A011635

Adjacent sequences: A229999 A230000 A230001 * A230003 A230004 A230005

Keyword

tabf,sign,easy

Author

Clark Kimberling, Nov 07 2013

Status

approved