OFFSET
0,2
COMMENTS
A transform of the Fibonacci numbers: The g.f. is the image of the g.f. x/(1-x-x^2) of A000045 under the transformation G(x) -> 1/(2*(1+x)) * G((1-x)/(2*(1+x)).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-10,-5).
FORMULA
G.f.: (1-x)/(1 + 10*x + 5*x^2).
a(n) = (-5 - 2*sqrt(5))^n*(1/2 + 3*sqrt(5)/10) + (-5 + 2*sqrt(5))^n*(1/2 - 3*sqrt(5)/10).
a(n) = (1/2)*5^((n-1)/2)*(sqrt(5)*(1+(-1)^n)*Fibonacci(3*n+2) - (1-(-1)^n)*Lucas(3*n+2)). - G. C. Greubel, Apr 21 2023
MATHEMATICA
LinearRecurrence[{-10, -5}, {1, -11}, 41] (* G. C. Greubel, Apr 21 2023 *)
PROG
(Magma) [n le 2 select 13-12*n else -10*Self(n-1) -5*Self(n-2): n in [1..41]]; // G. C. Greubel, Apr 21 2023
(SageMath)
def A099839(n): return 5^((n-1)/2)*(sqrt(5)*((n+1)%2)*fibonacci(3*n+2) - (n%2)*lucas_number2(3*n+2, 1, -1))
[A099839(n) for n in range(41)] # G. C. Greubel, Apr 21 2023
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Paul Barry, Oct 27 2004
STATUS
approved