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A111721 a(n) = a(n-1) + a(n-2) + 5 where a(0) = a(1) = 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 (list; graph; refs; listen; history; text; internal format)
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

Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences. [Wolfdieter Lang, Oct 17 2010]

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

a(n) = 3/2^n*((1+sqrt(5))^n + (1-sqrt(5))^n) + 3/(sqrt(5)*2^n)*((1+sqrt(5))^n - (1-sqrt(5))^n) - 5. - Paolo P. Lava, Jul 27 2011

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: A200270 A031887 A294943 * 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

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Last modified April 14 15:12 EDT 2021. Contains 342949 sequences. (Running on oeis4.)