This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A022086 #65 Apr 10 2025 06:51:59
%S A022086 0,3,3,6,9,15,24,39,63,102,165,267,432,699,1131,1830,2961,4791,7752,
%T A022086 12543,20295,32838,53133,85971,139104,225075,364179,589254,953433,
%U A022086 1542687,2496120,4038807,6534927,10573734,17108661,27682395,44791056,72473451,117264507
%N A022086 Fibonacci sequence beginning 0, 3.
%C A022086 First differences of A111314. - _Ross La Haye_, May 31 2006
%C A022086 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 A022086 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 A022086 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 7,17.
%H A022086 Vincenzo Librandi, <a href="/A022086/b022086.txt">Table of n, a(n) for n = 0..1000</a>
%H A022086 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A022086 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).
%F A022086 a(n) = 3*Fibonacci(n).
%F A022086 a(n) = F(n-2) + F(n+2) for n>1, with F=A000045.
%F A022086 a(n) = round( ((6*phi-3)/5) * phi^n ) for n>2. - _Thomas Baruchel_, Sep 08 2004
%F A022086 a(n) = A119457(n+1,n-1) for n>1. - _Reinhard Zumkeller_, May 20 2006
%F A022086 G.f.: 3*x/(1-x-x^2). - _Philippe Deléham_, Nov 19 2008
%F A022086 a(n) = A187893(n) - 1. - _Filip Zaludek_, Oct 29 2016
%F A022086 E.g.f.: 6*sinh(sqrt(5)*x/2)*exp(x/2)/sqrt(5). - _Ilya Gutkovskiy_, Oct 29 2016
%F A022086 a(n) = F(n+4) + F(n-4) - 4*F(n), F = A000045. - _Bruno Berselli_, Dec 29 2016
%p A022086 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 A022086 with (combinat):seq(sum((fibonacci(n,1)),m=1..3),n=0..32); # _Zerinvary Lajos_, Jun 19 2008
%t A022086 LinearRecurrence[{1, 1}, {0, 3}, 40] (* _Arkadiusz Wesolowski_, Aug 17 2012 *)
%t A022086 Table[Fibonacci[n + 4] + Fibonacci[n - 4] - 4 Fibonacci[n], {n, 0, 40}] (* _Bruno Berselli_, Dec 30 2016 *)
%t A022086 Table[3 Fibonacci[n], {n, 0, 40}] (* _Vincenzo Librandi_, Dec 31 2016 *)
%o A022086 (PARI) a(n)=3*fibonacci(n) \\ _Charles R Greathouse IV_, Nov 06 2014
%o A022086 (Magma) [3*Fibonacci(n): n in [0..40]]; // _Vincenzo Librandi_, Dec 31 2016
%o A022086 (SageMath)
%o A022086 def A022086(n): return 3*fibonacci(n)
%o A022086 print([A022086(n) for n in range(41)]) # _G. C. Greubel_, Apr 10 2025
%Y A022086 Essentially the same as A097135.
%Y A022086 Cf. A026390, A036999, A111314, A119457, A187893.
%Y A022086 Sequences of the form Fibonacci(n+k) + Fibonacci(n-k) are listed in A280154.
%Y A022086 Sequences of the form m*Fibonacci: A000045 (m=1), A006355 (m=2), this sequence (m=3), A022087 (m=4), A022088 (m=5), A022089 (m=6), A022090 (m=7), A022091 (m=8), A022092 (m=8), A022093 (m=10), A022345...A022366 (m=11...32).
%K A022086 nonn,easy
%O A022086 0,2
%A A022086 _N. J. A. Sloane_