A192384 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.

0, 2, 4, 24, 72, 312, 1088, 4288, 15744, 60192, 224832, 851072, 3197056, 12062592, 45398016, 171104256, 644354048, 2427699712, 9144222720, 34448209920, 129761986560, 488821962752, 1841370087424, 6936475090944, 26129575084032

Offset

1,2

Comments

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

The polynomial p(n,x) is defined by ((x+d)^n - (x-d)^n)/(2*d), where d = sqrt(x+3). 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 (2,8,-4,-4).

Formula

From Colin Barker, Dec 09 2012: (Start)

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

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

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

T(n, k) = [x^k] ((x+sqrt(x+3))^n - (x-sqrt(x+3))^n)/(2*sqrt(x+3)).

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 -> 6 + 4*x
  p(3, x) =    12*x +  4*x^2 +  4*x^3 -> 8 + 24*x
  p(4, x) = 9 + 6*x + 31*x^2 + 10*x^3 + 5*x^4 -> 60 + 72*x.
From these, read A192383 = (1, 0, 6, 8, 60, ...) and A192384 = (0, 2, 4, 24, 72, ...).

Mathematica

(See A192383.)
LinearRecurrence[{2, 8, -4, -4}, {0, 2, 4, 24}, 40] (* G. C. Greubel, Jul 10 2023 *)

Prog

(Magma)
R<x>:=PowerSeriesRing(Integers(), 41);
[0] cat Coefficients(R!( 2*x^2/(1-2*x-8*x^2+4*x^3+4*x^4) )) // G. C. Greubel, Jul 10 2023
(SageMath)
@CachedFunction
def a(n): # a = A192384
    if (n<5): return (0, 0, 2, 4, 24)[n]
    else: return 2*a(n-1) +8*a(n-2) -4*a(n-3) -4*a(n-4)
[a(n) for n in range(1, 41)] # G. C. Greubel, Jul 10 2023

Crossrefs

Cf. A192232, A192383, A192385.

Sequence in context: A068506 A272640 A192513 * A119036 A192382 A232205

Adjacent sequences: A192381 A192382 A192383 * A192385 A192386 A192387

Keyword

nonn,easy

Author

Clark Kimberling, Jun 30 2011

Status

approved