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 A022087 #65 Jul 31 2025 15:58:00
%S A022087 0,4,4,8,12,20,32,52,84,136,220,356,576,932,1508,2440,3948,6388,10336,
%T A022087 16724,27060,43784,70844,114628,185472,300100,485572,785672,1271244,
%U A022087 2056916,3328160,5385076,8713236,14098312,22811548,36909860,59721408,96631268
%N A022087 Fibonacci sequence beginning 0, 4.
%C A022087 For n > 1, this sequence gives the number of binary strings of length n that do not contain 0000, 0101, 1010, or 1111 as contiguous substrings (see A230127). - _Nathaniel Johnston_, Oct 11 2013
%D A022087 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 18.
%H A022087 Vincenzo Librandi, <a href="/A022087/b022087.txt">Table of n, a(n) for n = 0..1000</a>
%H A022087 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A022087 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).
%F A022087 a(n) = 4*F(n) = F(n-2) + F(n) + F(n+2), where F = A000045.
%F A022087 a(n) = round( phi^n*(8*phi-4)/5 ) for n>2. - _Thomas Baruchel_, Sep 08 2004
%F A022087 a(n) = A119457(n+2,n-1) for n>1. - _Reinhard Zumkeller_, May 20 2006
%F A022087 G.f.: 4*x/(1-x-x^2). - _Philippe Deléham_, Nov 19 2008
%F A022087 a(n) = F(n+9) - 17*F(n+3), where F=A000045. - _Manuel Valdivia_, Dec 15 2009
%F A022087 G.f.: Q(0) -1, where Q(k) = 1 + x^2 + (4*k+5)*x - x*(4*k+1 + x)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Oct 07 2013
%F A022087 a(n) = Fibonacci(n+3) - Fibonacci(n-3), where Fibonacci(-3..-1) = 2,-1,1. - _Bruno Berselli_, May 22 2015
%p A022087 a:= n-> (Matrix([[4,0]]). Matrix([[1,1],[1,0]])^n)[1,2]: seq(a(n), n=0..40); # _Alois P. Heinz_, Aug 17 2008
%t A022087 4*Fibonacci[Range[0,50]] (* _Vladimir Joseph Stephan Orlovsky_, Jul 22 2008 *)
%t A022087 Table[4 Fibonacci(n), {n, 0, 40}] (* _Bruno Berselli_, May 22 2015 *)
%t A022087 LinearRecurrence[{1,1},{0,4},40] (* _Harvey P. Dale_, Jul 31 2025 *)
%o A022087 (PARI) a(n)=4*fibonacci(n) \\ _Charles R Greathouse IV_, Jun 05 2011
%o A022087 (Magma) [4*Fibonacci(n): n in [0..40]]; // _Vincenzo Librandi_, Oct 12 2013
%o A022087 (SageMath)
%o A022087 def A022087(n): return 4*fibonacci(n)
%o A022087 print([A022087(n) for n in range(41)]) # _G. C. Greubel_, Apr 12 2025
%Y A022087 Cf. A000045, A119457.
%Y A022087 Cf. similar sequences listed in A258160.
%Y A022087 Cf. sequences of the form m*Fibonacci listed in A022086.
%K A022087 nonn,easy
%O A022087 0,2
%A A022087 _N. J. A. Sloane_