OFFSET
0,3
COMMENTS
This is the sequence A(0,1;1,1;9) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - Wolfdieter Lang, Oct 18 2010
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences.
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).
FORMULA
a(n) = a(n-1) + a(n-2) + 9 with a(0)=0 and a(1)=1.
From Wolfdieter Lang, Oct 18 2010: (Start)
O.g.f.: x*(1+8*x)/((1-x)*(1-x-x^2)).
a(n) = 2*a(n-1) - a(n-3), a(0)=0, a(1)=1, a(2)=10 (Observation by G. Detlefs).
(End)
a(n+1) - a(n) = A022099(n). - R. J. Mathar, Apr 22 2013
a(n) = -9 + ( (11 + 9*sqrt(5))*(1 + sqrt(5))^n - (11 - 9*sqrt(5))*(1 - sqrt(5))^n )/(2^(n+1)*sqrt(5)). - Colin Barker, Jul 13 2017
a(n) = Fibonacci(n+3) + 7*Fibonacci(n+1) - 9. - G. C. Greubel, Apr 25 2022
MATHEMATICA
CoefficientList[Series[x*(1+8*x)/((1-x)*(1-x-x^2)), {x, 0, 50}], x] (* G. C. Greubel, Jul 13 2017 *)
PROG
(PARI) concat(0, Vec(x*(1+8*x)/((1-x)*(1-x-x^2)) + O(x^50))) \\ Colin Barker, Jul 13 2017
(Magma) [Lucas(n+2) +6*Fibonacci(n+1) -9: n in [0..50]]; // G. C. Greubel, Apr 25 2022
(SageMath) [fibonacci(n+3) +7*fibonacci(n+1) -9 for n in (0..50)] # G. C. Greubel, Apr 25 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Mark Dols, Jan 28 2010
EXTENSIONS
Wrong offset 1 changed into 0 Wolfdieter Lang, Oct 18 2010
STATUS
approved