A111721 a(n) = a(n-1) + a(n-2) + 5 where a(0) = a(1) = 1.

1, 1, 7, 13, 25, 43, 73, 121, 199, 325, 529, 859, 1393, 2257, 3655, 5917, 9577, 15499, 25081, 40585, 65671, 106261, 171937, 278203, 450145, 728353, 1178503, 1906861, 3085369, 4992235, 8077609, 13069849, 21147463, 34217317, 55364785, 89582107

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

Cf. A000045, A000071.

Sequence in context: A031887 A294943 A363080 * A213663 A060455 A205541

Adjacent sequences: A111718 A111719 A111720 * A111722 A111723 A111724

Keyword

nonn,easy

Author

Parthasarathy Nambi, Nov 17 2005

Extensions

More terms from Stefan Steinerberger, Nov 19 2005

Status

approved