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A022086 Fibonacci sequence beginning 0, 3. 23
0, 3, 3, 6, 9, 15, 24, 39, 63, 102, 165, 267, 432, 699, 1131, 1830, 2961, 4791, 7752, 12543, 20295, 32838, 53133, 85971, 139104, 225075, 364179, 589254, 953433, 1542687, 2496120, 4038807, 6534927, 10573734, 17108661, 27682395, 44791056, 72473451, 117264507 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

First differences of A111314. - Ross La Haye, May 31 2006

Pisano period lengths: 1, 3, 1, 6, 20, 3, 16, 12, 8, 60, 10, 6, 28, 48, 20, 24, 36, 24, 18, 60, ... . - R. J. Mathar, Aug 10 2012

REFERENCES

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 7,17.

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

a(n) = round( ((6*phi-3)/5) * phi^n ) for n>2. - Thomas Baruchel, Sep 08 2004

a(n) = 3*F(n). Also, a(n) = F(n-2) + F(n+2) for n>1, with F=A000045.

a(n) = A119457(n+1,n-1) for n>1. - Reinhard Zumkeller, May 20 2006

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

a(n) = A187893(n) - 1. - Filip Zaludek, Oct 29 2016

E.g.f.: 6*sinh(sqrt(5)*x/2)*exp(x/2)/sqrt(5). - Ilya Gutkovskiy, Oct 29 2016

a(n) = F(n+4) + F(n-4) - 4*F(n). - Bruno Berselli, Dec 29 2016

MAPLE

BB := n->if n=0 then 3; > elif n=1 then 0; > else BB(n-2)+BB(n-1); > fi: > L:=[]: for k from 1 to 34 do L:=[op(L), BB(k)]: od: L; # Zerinvary Lajos, Mar 19 2007

with (combinat):seq(sum((fibonacci(n, 1)), m=1..3), n=0..32); # Zerinvary Lajos, Jun 19 2008

MATHEMATICA

LinearRecurrence[{1, 1}, {0, 3}, 40] (* Arkadiusz Wesolowski, Aug 17 2012 *)

Table[Fibonacci[n + 4] + Fibonacci[n - 4] - 4 Fibonacci[n], {n, 0, 40}] (* Bruno Berselli, Dec 30 2016 *)

Table[3 Fibonacci[n], {n, 0, 40}] (* Vincenzo Librandi, Dec 31 2016 *)

PROG

(PARI) a(n)=3*fibonacci(n) \\ Charles R Greathouse IV, Nov 06 2014

(MAGMA) [3*Fibonacci(n): n in [0..40]]; // Vincenzo Librandi, Dec 31 2016

CROSSREFS

Essentially the same as A097135. Cf. A026390, A036999.

Cf. sequences with formula Fibonacci(n+k)+Fibonacci(n-k) listed in A280154.

Sequence in context: A035528 A050337 * A097135 A274498 A167786 A167787

Adjacent sequences:  A022083 A022084 A022085 * A022087 A022088 A022089

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified March 25 09:42 EDT 2017. Contains 284060 sequences.