A022136 Fibonacci sequence beginning 5, 11.

5, 11, 16, 27, 43, 70, 113, 183, 296, 479, 775, 1254, 2029, 3283, 5312, 8595, 13907, 22502, 36409, 58911, 95320, 154231, 249551, 403782, 653333, 1057115, 1710448, 2767563, 4478011, 7245574, 11723585, 18969159, 30692744, 49661903, 80354647, 130016550

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

From Wajdi Maaloul, Jun 19 2022: (Start)

For n > 0, 2*a(n) is the number of ways to tile this strip of length n+1 (beginning with two vertical strips of length 3) using squares and dominoes.

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From Greg Dresden, Jun 26 2022: (Start)

For n > 0, 5*a(n) is the number of ways to tile this strip of length n (beginning with a long vertical strip of length 9) using squares and dominoes.

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(End)

The 5-adic evaluation of this sequence is A112765(a(n)) = 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2,.. and has records of nu_5 = 1, 2, 3, 4, 5, 7, 8,... at n=0, 10, 85, 585, 3085, 12460, 168710,... Does this grow without bounds? [StackExchange] - R. J. Mathar, Jun 28 2025

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Tanya Khovanova, Recursive Sequences
• StackExchange, nu_5(5F_(n+2)+F_n), Jun 2025.
• Index entries for linear recurrences with constant coefficients, signature (1,1).

Crossrefs

Cf. A000032, A000045.

Programs

Mathematica

Transpose[NestList[{Last[#],Total[#]}&,{5,11},40]][[1]]  (* Harvey P. Dale, Apr 04 2011 *)
LinearRecurrence[{1, 1}, {5, 11}, 40] (* Vincenzo Librandi, May 03 2017 *)

Formula

G.f.: (5+6*x)/(1-x-x^2). - Philippe Deléham, Nov 20 2008

a(n) = 2*Lucas(n-1) + 7*Fibonacci(n+1). - Lechoslaw Ratajczak, May 01 2017

a(n) = Fibonacci(n+6) - Fibonacci(n+1) - Lucas(n). - Greg Dresden and Aamen Muharram, Jul 22 2022