A022086 - OEIS

%I #60 Oct 23 2024 01:25:04

%S 0,3,3,6,9,15,24,39,63,102,165,267,432,699,1131,1830,2961,4791,7752,

%T 12543,20295,32838,53133,85971,139104,225075,364179,589254,953433,

%U 1542687,2496120,4038807,6534927,10573734,17108661,27682395,44791056,72473451,117264507

%N Fibonacci sequence beginning 0, 3.

%C First differences of A111314. - _Ross La Haye_, May 31 2006

%C 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

%C For n>=6, a(n) is the number of edge covers of the union of two cycles C_r and C_s, r+s=n, with a single common vertex. - _Feryal Alayont_, Oct 17 2024

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

%H Vincenzo Librandi, <a href="/A022086/b022086.txt">Table of n, a(n) for n = 0..1000</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1).

%F a(n) = round( ((6*phi-3)/5) * phi^n ) for n>2. - _Thomas Baruchel_, Sep 08 2004

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

%F a(n) = A119457(n+1,n-1) for n>1. - _Reinhard Zumkeller_, May 20 2006

%F G.f.: 3*x/(1-x-x^2). - _Philippe Deléham_, Nov 19 2008

%F a(n) = A187893(n) - 1. - _Filip Zaludek_, Oct 29 2016

%F E.g.f.: 6*sinh(sqrt(5)*x/2)*exp(x/2)/sqrt(5). - _Ilya Gutkovskiy_, Oct 29 2016

%F a(n) = F(n+4) + F(n-4) - 4*F(n). - _Bruno Berselli_, Dec 29 2016

%p 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

%p with (combinat):seq(sum((fibonacci(n,1)),m=1..3),n=0..32); # _Zerinvary Lajos_, Jun 19 2008

%t LinearRecurrence[{1, 1}, {0, 3}, 40] (* _Arkadiusz Wesolowski_, Aug 17 2012 *)

%t Table[Fibonacci[n + 4] + Fibonacci[n - 4] - 4 Fibonacci[n], {n, 0, 40}] (* _Bruno Berselli_, Dec 30 2016 *)

%t Table[3 Fibonacci[n], {n, 0, 40}] (* _Vincenzo Librandi_, Dec 31 2016 *)

%o (PARI) a(n)=3*fibonacci(n) \\ _Charles R Greathouse IV_, Nov 06 2014

%o (Magma) [3*Fibonacci(n): n in [0..40]]; // _Vincenzo Librandi_, Dec 31 2016

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

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

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_