A192387 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.

0, 2, 4, 32, 96, 512, 1856, 8576, 33792, 147456, 602112, 2566144, 10637312, 44892160, 187269120, 787087360, 3292069888, 13812760576, 57837355008, 242497880064, 1015868817408, 4258009186304, 17841063460864, 74771320537088, 313317428035584

Offset

1,2

Comments

The polynomial p(n,x) is defined by ((x+d)^n - (x-d)^n)/(2*d), where d = sqrt(x+5). 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 = 1..1000

Index entries for linear recurrences with constant coefficients, signature (2,12,-8,-16).

Formula

From Colin Barker, Dec 09 2012: (Start)

a(n) = 2*a(n-1) + 12*a(n-2) - 8*a(n-3) - 16*a(n-4).

G.f.: 2*x^2/(1-2*x-12*x^2+8*x^3+16*x^4). (End)

a(n) = 2^n*A112576(n). - R. J. Mathar, Mar 08 2021

From G. C. Greubel, Jul 10 2023: (Start)

T(n, k) = [x^k] ((x+sqrt(x+5))^n - (x-sqrt(x+5))^n)/(2*sqrt(x+5)).

a(n) = Sum_{k=0..n-1} T(n, k)*Fibonacci(k). (End)

Example

The first five polynomials p(n,x) and their reductions are as follows:
  p(0,x) = 1 -> 1
  p(1,x) =     2*x -> 2*x
  p(2,x) = 3 +   x +  3*x^2 -> 8 + 4*x
  p(3,x) =    12*x +  4*x^2 +  4*x^3 -> 8 + 32*x
  p(4,x) = 9 + 6*x + 31*x^2 + 10*x^3 + 5*x^4 -> 96 + 96*x.
From these, read A192386 = (1, 0, 8, 8, 96, ...) and a(n) = (0, 2, 4, 32, 96, ...).

Mathematica

(See A192386.)
LinearRecurrence[{2, 12, -8, -16}, {0, 2, 4, 32}, 40] (* G. C. Greubel, Jul 10 2023 *)

Prog

(Magma)
R<x>:=PowerSeriesRing(Integers(), 41);
[0] cat Coefficients(R!( 2*x^2/(1-2*x-12*x^2+8*x^3+16*x^4) )); // G. C. Greubel, Jul 10 2023
(SageMath)
@CachedFunction
def a(n): # a = A192387
    if (n<5): return (0, 0, 2, 4, 32)[n]
    else: return 2*a(n-1) +12*a(n-2) -8*a(n-3) -16*a(n-4)
[a(n) for n in range(1, 41)] # G. C. Greubel, Jul 10 2023

Crossrefs

Cf. A112576, A192232, A192386, A192388.

Sequence in context: A101575 A197099 A009098 * A320624 A133127 A122693

Adjacent sequences: A192384 A192385 A192386 * A192388 A192389 A192390

Keyword

nonn,easy

Author

Clark Kimberling, Jun 30 2011

Status

approved