A022120 Fibonacci sequence beginning 3, 7.

3, 7, 10, 17, 27, 44, 71, 115, 186, 301, 487, 788, 1275, 2063, 3338, 5401, 8739, 14140, 22879, 37019, 59898, 96917, 156815, 253732, 410547, 664279, 1074826, 1739105, 2813931, 4553036, 7366967, 11920003, 19286970, 31206973, 50493943, 81700916, 132194859

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

From Greg Dresden, Feb 18 2022: (Start)

a(n) is also the number of ways to tile this figure, with two cells on the top row and n+1 cells on the bottom row, using squares and dominoes. Shown here are the figures for a(0) through a(4):

._ ._ ._ ._ ._

||| ||| |||_ |||___ |||_____

|| ||_| |||_| |||_|| ||_|||_|

(End)

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..4782
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (1,1).

Crossrefs

Cf. A000032.

Programs

Mathematica

Table[4*Fibonacci[n+2]-Fibonacci[n+1],{n,0,30}] (* Zak Seidov, Mar 15 2011 *)
LinearRecurrence[{1,1},{3,7},50] (* Harvey P. Dale, May 28 2024 *)

PARI

v=vector(100);v[1]=3;v[2]=7;for(i=3,#v,v[i]=v[i-2]+v[i-1]);v \\ Charles R Greathouse IV, Mar 15 2011

Formula

G.f.: (3+4x)/(1-x-x^2). - Philippe Deléham, Nov 19 2008

a(n) = 4*Fibonacci(n+2) - Fibonacci(n+1). - Gary Detlefs, Dec 21 2010

a(n) = round(((15+11*sqrt(5))/10)*((1+sqrt(5))/2)^n + ((15-11*sqrt(5))/10)*((1-sqrt(5))/2)^n). - Bogart B. Strauss, Oct 27 2013

a(n) = Lucas(n+3) - Fibonacci(n-1). - Greg Dresden, Sam Neale, and Kyle Wood, Feb 18 2022

E.g.f.: exp(x/2)*(15*cosh(sqrt(5)*x/2) + 11*sqrt(5)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, Jul 26 2022