A192388 a(n) = A192387(n)/2.

0, 1, 2, 16, 48, 256, 928, 4288, 16896, 73728, 301056, 1283072, 5318656, 22446080, 93634560, 393543680, 1646034944, 6906380288, 28918677504, 121248940032, 507934408704, 2129004593152, 8920531730432, 37385660268544, 156658714017792

Offset

1,3

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 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) = (1/2)*Sum_{k=0..n-1} T(n, k)*Fibonacci(k-1).

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

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

a(n) = 2^(n-1)*A112576(n). (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 a(n) = (0, 2, 4, 32, 96, ...)/2 = (0, 1, 2, 16, 48, ...).

Mathematica

(* See A192386 *)
LinearRecurrence[{2, 12, -8, -16}, {0, 1, 2, 16}, 40] (* G. C. Greubel, Jul 10 2023 *)

Prog

(Magma)
R<x>:=PowerSeriesRing(Integers(), 41);
[0] cat Coefficients(R!( 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 = A192388
    if (n<5): return (0, 0, 1, 2, 16)[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, A192387.

Sequence in context: A220173 A220801 A220250 * A162442 A159010 A223219

Adjacent sequences: A192385 A192386 A192387 * A192389 A192390 A192391

Keyword

nonn,easy

Author

Clark Kimberling, Jun 30 2011

Status

approved