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%I M2362 #40 Sep 08 2022 08:44:34
%S 1,0,3,4,10,18,35,64,117,210,374,660,1157,2016,3495,6032,10370,17766,
%T 30343,51680,87801,148830,251758,425064,716425,1205568,2025675,
%U 3399004,5696122,9534330,15941099,26625280,44426877,74062506,123360230,205303932,341416205,567353376
%N a(1) = 1, a(2) = 0; for n > 2, a(n) = n*Fibonacci(n-2) (with the convention Fibonacci(0)=0, Fibonacci(1)=1).
%C Number of circular binary words of length n having exactly one occurrence of 00. Example: a(5)=10 because we have 00111, 10011, 11001, 11100, 01110, 00101, 10010, 01001, 10100 and 01010. Column 1 of A119458. - _Emeric Deutsch_, May 20 2006
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A006490/b006490.txt">Table of n, a(n) for n = 1..1000</a>
%H Ricardo Gómez Aíza, <a href="https://arxiv.org/abs/2009.02669">Symbolic dynamical scales: modes, orbitals, and transversals</a>, arXiv:2009.02669 [math.DS], 2020.
%H L. Carlitz and R. Scoville, <a href="http://www.fq.math.ca/Scanned/15-3/carlitz1.pdf">Zero-one sequences and Fibonacci numbers</a>, Fibonacci Quarterly, 15 (1977), 246-254.
%H J. P. McSorley, <a href="http://dx.doi.org/10.1016/S0012-365X(97)00086-1">Counting structures in the Moebius ladder</a>, Discrete Math., 184 (1998), 137-164.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2,-1).
%F G.f.: x(1-2x+2x^2)/(1-x-x^2)^2. - _Emeric Deutsch_, May 20 2006
%F a(n) = 2*a(n-1)+a(n-2)-2*a(n-3)-a(n-4). - _Vincenzo Librandi_, Aug 07 2017
%p with(combinat): a[1]:=1: a[2]:=0: for n from 3 to 40 do a[n]:=n*fibonacci(n-2) od: seq(a[n],n=1..40); # _Emeric Deutsch_, May 20 2006
%p A006490:=(1-2*z+2*z**2)/(z**2+z-1)**2; # conjectured by _Simon Plouffe_ in his 1992 dissertation
%t Table[Sum[Fibonacci[n - 1], {i, 0, n}], {n, 0, 34}] (* _Zerinvary Lajos_, Jul 12 2009 *)
%t CoefficientList[Series[(1 - 2 x + 2 x^2) / (1 - x - x^2)^2, {x, 0, 33}], x] (* or *) LinearRecurrence[{2, 1, -2, -1}, {1, 0, 3, 4}, 40] (* _Vincenzo Librandi_, Aug 07 2017 *)
%o (Magma) [n*Fibonacci(n-2): n in [1..40]]; // _Vincenzo Librandi_, Aug 07 2017
%o (PARI) a(n) = n*fibonacci(n-2); \\ _Michel Marcus_, Aug 07 2017
%Y Cf. A000045, A119458.
%K nonn
%O 1,3
%A _N. J. A. Sloane_
%E Better definition from _Ralf Stephan_, Nov 18 2004
%E More terms from _Emeric Deutsch_, May 20 2006
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