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%I #56 Sep 08 2022 08:44:45
%S 0,4,4,8,12,20,32,52,84,136,220,356,576,932,1508,2440,3948,6388,10336,
%T 16724,27060,43784,70844,114628,185472,300100,485572,785672,1271244,
%U 2056916,3328160,5385076,8713236,14098312,22811548,36909860,59721408,96631268
%N Fibonacci sequence beginning 0, 4.
%C 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 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 18.
%H Vincenzo Librandi, <a href="/A022087/b022087.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) = 4*F(n) = F(n-2) + F(n) + F(n+2), where F = A000045.
%F a(n) = round( phi^n*(8*phi-4)/5 ) for n>2. - _Thomas Baruchel_, Sep 08 2004
%F a(n) = A119457(n+2,n-1) for n>1. - _Reinhard Zumkeller_, May 20 2006
%F G.f.: 4*x/(1-x-x^2). - _Philippe Deléham_, Nov 19 2008
%F a(n) = F(n+9) - 17*F(n+3), where F=A000045. - _Manuel Valdivia_, Dec 15 2009
%F 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 a(n) = Fibonacci(n+3) - Fibonacci(n-3), where Fibonacci(-3..-1) = 2,-1,1. [_Bruno Berselli_, May 22 2015]
%p 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 a={};b=0;c=4;AppendTo[a,b];AppendTo[a,c];Do[b=b+c;AppendTo[a,b];c=b+c;AppendTo[a,c],{n,1,9,1}];a (* _Vladimir Joseph Stephan Orlovsky_, Jul 22 2008 *)
%t Table[4 Fibonacci(n), {n, 0, 40}] (* _Bruno Berselli_, May 22 2015 *)
%o (PARI) a(n)=4*fibonacci(n) \\ _Charles R Greathouse IV_, Jun 05 2011
%o (Magma) [4*Fibonacci(n): n in [0..40]]; // _Vincenzo Librandi_, Oct 12 2013
%Y Cf. A000045.
%Y Cf. similar sequences listed in A258160.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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