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%I #57 Sep 08 2022 08:44:45
%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
%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_
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