A022121 Fibonacci sequence beginning 3, 8.

3, 8, 11, 19, 30, 49, 79, 128, 207, 335, 542, 877, 1419, 2296, 3715, 6011, 9726, 15737, 25463, 41200, 66663, 107863, 174526, 282389, 456915, 739304, 1196219, 1935523, 3131742, 5067265, 8199007, 13266272, 21465279, 34731551, 56196830, 90928381, 147125211

Offset

0,1

Comments

From Wajdi Maaloul, Jun 20 2022: (Start)

a(n) is the number of ways to tile the figure below with squares and dominoes (a strip of length n+2 that contains two vertical strip of height 2 in its first and third tiles).

_ _

|_|_|_|_____ _

|_|_|_|_|_|_|...|_|

(End)

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (1,1).

Formula

G.f.: (3+5*x)/(1-x-x^2). - Philippe Deléham, Nov 19 2008

a(n) = F(n+2) + 2*L(n+1), where F(n) and L(n) are the n-th Fibonacci and Lucas number, respectively. - Rigoberto Florez, Jan 29 2020

E.g.f.: exp(-2*x/(1+sqrt(5)))*(-2*(5+6*sqrt(5))+(55+27*sqrt(5))*exp(sqrt(5)*x))/(5*(3+sqrt(5))). - Stefano Spezia, Jan 29 2020 after Rigoberto Florez

a(n) = Lucas(n+3) + Fibonacci(n-2). - Greg Dresden and Steve Warren, Feb 28 2022

Mathematica

LinearRecurrence[{1, 1}, {3, 8}, 40] (* Vladimir Joseph Stephan Orlovsky, Sep 17 2008 *)
Table[2 LucasL[n+1] + Fibonacci[n+2], {n, 0, 40}] (* Rigoberto Florez, Jan 29 2020 *)

Prog

(Magma) a0:=3; a1:=8; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..40]]; // Vincenzo Librandi, Jan 25 2017
(PARI) apply( {A022121(n)=[3, 8]*([0, 1; 1, 1]^n)[, 1]}, [0..30]) \\ M. F. Hasler, Feb 28 2020

Crossrefs

Cf. A000032, A000045.

Sequence in context: A105410 A114548 A182759 * A364086 A171672 A341262

Adjacent sequences: A022118 A022119 A022120 * A022122 A022123 A022124

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved