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 (2,8,-4,-4).
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