A192385 a(n) = A192384(n)/2.

0, 1, 2, 12, 36, 156, 544, 2144, 7872, 30096, 112416, 425536, 1598528, 6031296, 22699008, 85552128, 322177024, 1213849856, 4572111360, 17224104960, 64880993280, 244410981376, 920685043712, 3468237545472, 13064787542016

Offset: 1

Author

Clark Kimberling, Jun 30 2011

Keywords

• nonn

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).

Crossrefs

Cf. A192232, A192383, A192384.

Programs

Magma

R:=PowerSeriesRing(Integers(), 41);
[0] cat Coefficients(R!( x^2/(1-2*x-8*x^2+4*x^3+4*x^4) )); // G. C. Greubel, Jul 10 2023

Mathematica

(See A192384.)
LinearRecurrence[{2,8,-4,-4}, {0,1,2,12}, 40] (* G. C. Greubel, Jul 10 2023 *)

SageMath

@CachedFunction
def a(n): # a = A192385
    if (n<5): return (0,0,1,2,12)[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

Formula

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

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

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