OFFSET
0,3
COMMENTS
a(n+1)/a(n) converges to the golden ratio. - Stefan Steinerberger, Nov 19 2005
This is the sequence A(1,1;1,1;5) of the family of sequences [a,b:c,d:k] considered by Gary Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - Wolfdieter Lang, Oct 17 2010
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries, Ann. Math. Silesianae (2023). See p. 13.
Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences.
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).
FORMULA
For n > 1: a(n) = a(n-1) + 6*F(n-1). (a(n)-1)/6 = A000071(n+1) = F(n+1) - 1. Hence a(n) = 6*F(n+1) - 5. - Jonathan Vos Post, Nov 19 2005
G.f.: (5*x^2-x+1)/(x^3-2*x+1). - Stefan Steinerberger, Nov 19 2005
EXAMPLE
a(2) = a(0) + a(1) + 5 = 1 + 1 + 5 = 7.
MATHEMATICA
Join[{a=1, b=1}, Table[c=a+b+5; a=b; b=c, {n, 50}]] (* Vladimir Joseph Stephan Orlovsky, Apr 13 2011 *)
Nest[Append[#, #[[-1]] + #[[-2]] + 5] &, {1, 1}, 34] (* or *)
CoefficientList[Series[(5 x^2 - x + 1)/(x^3 - 2 x + 1), {x, 0, 35}], x] (* Michael De Vlieger, Dec 17 2017 *)
PROG
(MuPAD) a := 1; b := 1; for n from 1 to 50 do c := a+b+5; print(c); a := b; b := c; end_for; // Stefan Steinerberger
(Sage) from sage.combinat.sloane_functions import recur_gen2b
it =recur_gen2b(1, 1, 1, 1, lambda n: 5)
[next(it) for i in range(38)] # Zerinvary Lajos, Jul 16 2008
(Haskell)
a111721 n = a111721_list !! n
a111721_list = 1 : 1 :
map (+ 5) (zipWith (+) a111721_list (tail a111721_list))
-- Reinhard Zumkeller, Nov 05 2011
(PARI) x='x+O('x^99); Vec((5*x^2-x+1)/(x^3-2*x+1)) \\ Altug Alkan, Dec 17 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Parthasarathy Nambi, Nov 17 2005
EXTENSIONS
More terms from Stefan Steinerberger, Nov 19 2005
STATUS
approved