OFFSET
1,2
COMMENTS
Limit_{n->infinity} a(n)/a(n-1) = phi.
LINKS
W. Kuszmaul, Fast Algorithms for Finding Pattern Avoiders and Counting Pattern Occurrences in Permutations, arXiv preprint arXiv:1509.08216 [cs.DM], 2015-2017.
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).
FORMULA
Given n-th Lucas number A000204(n), a(n) = 2*L(n) + L(n-1) - n.
G.f.: -x*(1-5*x^2+x^3+2*x+2*x^4)/(-1+x+x^2)/(-1+x)^2. - R. J. Mathar, Nov 14 2007
a(n) = A000032(n+2) - n = Fibonacci(n+3) + Fibonacci(n+1) - n, n > 1. - R. J. Mathar, Jul 20 2009, extended by David A. Corneth, Aug 08 2018
EXAMPLE
a(5) = 24 = 2*L(5) + L(4) - n = 2*11 + 7 - 5.
MATHEMATICA
a[1] = 1; a[n_] := LucasL[n+2] - n;
Array[a, 14] (* Jean-François Alcover, Aug 08 2018, after R. J. Mathar *)
PROG
(PARI) a(n) = {if(n==1, 1, fibonacci(n+3)+fibonacci(n+1)-n)} \\ David A. Corneth, Aug 08 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Gary W. Adamson, Sep 19 2007
STATUS
approved