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A192385
a(n) = A192384(n)/2.
3
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
1,3
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)
MATHEMATICA
(See A192384.)
LinearRecurrence[{2, 8, -4, -4}, {0, 1, 2, 12}, 40] (* G. C. Greubel, Jul 10 2023 *)
PROG
(Magma)
R<x>:=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
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jun 30 2011
STATUS
approved