A192426 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.

2, 0, 5, 1, 18, 13, 81, 106, 413, 729, 2258, 4653, 12833, 28666, 74493, 173545, 437346, 1041421, 2583089, 6221322, 15304541, 37079289, 90826994, 220729069, 539487297, 1313161498, 3205831869, 7809748489, 19054635650, 46439068365

Offset

0,1

Comments

The polynomial p(n,x) is defined by ((x+d)/2)^n + ((x-d)/2)^n, where d = sqrt(x^2+8). For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1,5,-2,-4).

Formula

From Colin Barker, May 12 2014: (Start)

a(n) = a(n-1) + 5*a(n-2) - 2*a(n-3) - 4*a(n-4).

G.f.: (2-2*x-5*x^2)/(1-x-5*x^2+2*x^3+4*x^4). (End)

a(n) = Sum_{k=0..n} T(n, k)*Fibonacci(k-1), where T(n, k) = [x^k] ((x + sqrt(x^2+8))^n + (x - sqrt(x^2+8))^n)/2^n. - G. C. Greubel, Jul 12 2023

Example

The first five polynomials p(n,x) and their reductions are as follows:
  p(0,x) = 2 -> 2
  p(1,x) = x -> x
  p(2,x) = 4 + x^2 -> 5 + x
  p(3,x) = 6*x + x^3 -> 1 + 8*x
  p(4,x) = 8 + 8*x^2 + x^4 -> 18 + 11*x.
From these, read a(n) = (2, 0, 5, 1, 18, ...) and A192427 = (0, 1, 1, 8, 11, ...).

Mathematica

q[x_]:= x+1; d= Sqrt[x^2+8];
p[n_, x_]:= ((x+d)/2)^n + ((x-d)/2)^n (* suggested by A162514 *)
Table[Expand[p[n, x]], {n, 0, 6}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};
t= Table[FixedPoint[Expand[#1/. reductionRules] &, p[n, x]], {n, 0, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 30}] (* A192426 *)
Table[Coefficient[Part[t, n], x, 1], {n, 30}] (* A192427 *)
LinearRecurrence[{1, 5, -2, -4}, {2, 0, 5, 1}, 40] (* G. C. Greubel, Jul 12 2023 *)

Prog

(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (2-2*x-5*x^2)/(1-x-5*x^2+2*x^3+4*x^4) )); // G. C. Greubel, Jul 12 2023
(SageMath)
@CachedFunction
def a(n): # a = A192426
    if (n<4): return (2, 0, 5, 1)[n]
    else: return a(n-1) + 5*a(n-2) - 2*a(n-3) - 4*a(n-4)
[a(n) for n in range(41)] # G. C. Greubel, Jul 12 2023

Crossrefs

Cf. A000045, A162514, A192232, A192427.

Sequence in context: A290395 A082974 A167635 * A075603 A264357 A221573

Adjacent sequences: A192423 A192424 A192425 * A192427 A192428 A192429

Keyword

nonn,easy

Author

Clark Kimberling, Jun 30 2011

Extensions

Typo in name corrected by G. C. Greubel, Jul 12 2023

Status

approved