A022130 Fibonacci sequence beginning 4,9.

4, 9, 13, 22, 35, 57, 92, 149, 241, 390, 631, 1021, 1652, 2673, 4325, 6998, 11323, 18321, 29644, 47965, 77609, 125574, 203183, 328757, 531940, 860697, 1392637, 2253334, 3645971, 5899305, 9545276, 15444581, 24989857, 40434438, 65424295, 105858733, 171283028

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

The associated Pisano series starts as in A001175, but differs for example for modulus 29 where it is 7, not 14. - R. J. Mathar, Nov 02 2011

The Pisano period also differs for modulus 58, where it is 21 instead of 42. Otherwise, the Pisano periods coincide with those of the Fibonacci numbers. - Klaus Purath, Jun 26 2022

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Tanya Khovanova, Recursive Sequences
• H. Zhao and X. Li, On the Fibonacci numbers of trees, Fib. Quart., 44 (2006), 32-38.
• Index entries for linear recurrences with constant coefficients, signature (1,1).

Crossrefs

Cf. A000032, A001175.

Programs

Magma

a0:=4; a1:=9; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..35]]; // Vincenzo Librandi, Jan 25 2017

Maple

a:= n-> (<<0|1>, <1|1>>^n.<<4, 9>>)[1,1]:
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 22 2017

Mathematica

a={};b=4;c=9;AppendTo[a,b];AppendTo[a,c];Do[b=b+c;AppendTo[a,b];c=b+c;AppendTo[a,c],{n,1,40,1}];a (* Vladimir Joseph Stephan Orlovsky, Jul 23 2008 *)
LinearRecurrence[{1,1},{4,9},40] (* Harvey P. Dale, Dec 15 2011 *)

PARI

a(n)=4*fibonacci(n-1)+9*fibonacci(n) \\ Charles R Greathouse IV, Jun 05 2011

Formula

a(n) = 4*Fibonacci(n+2) + Fibonacci(n).

G.f.: (4 + 5*x)/(1-x-x^2). - Philippe Deléham, Nov 19 2008

a(n)= Fibonacci(n-2) + Fibonacci(n+5). - Gary Detlefs, Mar 31 2012