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

Author

Clark Kimberling, Jun 30 2011

Keywords

• nonn
• easy

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.

Examples

			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, ...).
		

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).

Crossrefs

Cf. A112576, A192232, A192386, A192388.

Programs

Magma

R:=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

Mathematica

(See A192386.)
LinearRecurrence[{2,12,-8,-16}, {0,2,4,32}, 40] (* 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

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)