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%I #161 Dec 29 2023 11:35:52
%S 1,0,2,2,4,6,10,16,26,42,68,110,178,288,466,754,1220,1974,3194,5168,
%T 8362,13530,21892,35422,57314,92736,150050,242786,392836,635622,
%U 1028458,1664080,2692538,4356618,7049156,11405774,18454930,29860704,48315634
%N Number of binary vectors of length n containing no singletons.
%C Number of cvtemplates at n-2 letters given <= 2 consecutive consonants or vowels (n >= 4).
%C Number of (n,2) Freiman-Wyner sequences.
%C Diagonal sums of the Riordan array ((1-x+x^2)/(1-x), x/(1-x)), A072405 (where this begins 1,0,1,1,1,1,...). - _Paul Barry_, May 04 2005
%C Central terms of the triangle in A094570. - _Reinhard Zumkeller_, Mar 22 2011
%C Pisano period lengths: 1, 1, 8, 3, 20, 8, 16, 6, 24, 20, 10, 24, 28, 16, 40, 12, 36, 24, 18, 60, ... . - _R. J. Mathar_, Aug 10 2012
%C Also the number of matchings in the (n-2)-pan graph for n >= 5. - _Eric W. Weisstein_, Oct 03 2017
%C a(n) is the number of bimultus bitstrings of length n. A bitstring is bimultus if each of its 1's possess at least one neighboring 1 and each of its 0's possess at least one neighboring 0. - _Steven Finch_, May 26 2020
%D A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 16, 51.
%H Charles R Greathouse IV, <a href="/A006355/b006355.txt">Table of n, a(n) for n = 0..4786</a> (next term has 1001 digits)
%H Kassie Archer and Aaron Geary, <a href="https://arxiv.org/abs/2312.14351">Powers of permutations that avoid chains of patterns</a>, arXiv:2312.14351 [math.CO], 2023. See p. 15.
%H J. Berman and P. Koehler, <a href="/A006356/a006356.pdf">Cardinalities of finite distributive lattices</a>, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
%H Ian F. Blake, <a href="http://dx.doi.org/10.1016/S0019-9958(82)90591-5">The enumeration of certain run length sequences</a>, Information and Control, 55 (1982), 222-237.
%H A. Burstein, S. Kitaev, and T. Mansour, <a href="http://puma.dimai.unifi.it/19_2_3/3.pdf">Partially ordered patterns and their combinatorial interpretations</a>, PU. M. A. Vol. 19 (2008), No. 2-3, pp. 27-38.
%H Steven Finch, <a href="https://arxiv.org/abs/2005.12185">Variance of longest run duration in a random bitstring</a>, arXiv:2005.12185 [math.CO], 2020.
%H Enoch Haga, <a href="http://digitalcommons.butler.edu/wordways/vol33/iss2/7">Room for Expansion</a>, Word Ways, 33 (No. 2, 2000), pp. 106-113 (see p. 110).
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=898">Encyclopedia of Combinatorial Structures 898</a>.
%H Sergey Kitaev and Jeffrey Remmel, <a href="http://arxiv.org/abs/1304.4286">(a,b)-rectangle patterns in permutations and words</a>, arXiv:1304.4286 [math.CO], 2013.
%H Noriaki Sannomiya, H. Katsura, and Y. Nakayama, <a href="http://arxiv.org/abs/1612.02285">Supersymmetry breaking and Nambu-Goldstone fermions with cubic dispersion</a>, arXiv preprint arXiv:1612.02285 [cond-mat.str-el], 2016-2017. See Table II, line 2.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IndependentEdgeSet.html">Independent Edge Set</a>, <a href="http://mathworld.wolfram.com/Matching.html">Matching</a> and <a href="http://mathworld.wolfram.com/PanGraph.html">Pan Graph</a>.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).
%F a(n+2) = F(n-1) + F(n+2), for n > 0.
%F G.f.: (1-x+x^2)/(1-x-x^2). - _Paul Barry_, May 04 2005
%F a(n) = A119457(n-1,n-2) for n > 2. - _Reinhard Zumkeller_, May 20 2006
%F a(n) = 2*F(n-1) for n > 0, F(n)=A000045(n) and a(0)=1. - _Mircea Merca_, Jun 28 2012
%F G.f.: 1 - x + x*Q(0), where Q(k) = 1 + x^2 + (2*k+3)*x - x*(2*k+1 + x)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Oct 05 2013
%F a(n) = A118658(n) - 0^n. - _M. F. Hasler_, Nov 05 2014
%F a(n) = (2^(-n)*((1-r)^n*(1+r) + (-1+r)*(1+r)^n)) / r for n > 0, where r=sqrt(5). - _Colin Barker_, Jan 28 2017
%F a(n) = a(n-1) + a(n-2) for n >= 3. - _Armend Shabani_, Nov 25 2020
%F E.g.f.: 2*exp(x/2)*(5*cosh(sqrt(5)*x/2) - sqrt(5)*sinh(sqrt(5)*x/2))/5 - 1. - _Stefano Spezia_, Apr 18 2022
%e a(6)=10 because we have: 000000, 000011, 000111, 001100, 001111, 110000, 110011, 111000, 111100, 111111. - _Geoffrey Critzer_, Jan 26 2014
%p a:= n-> if n=0 then 1 else (Matrix([[2,-2]]). Matrix([[1,1], [1,0]])^n) [1,1] fi: seq(a(n), n=0..38); # _Alois P. Heinz_, Aug 18 2008
%p a := n -> ifelse(n=0, 1, -2*I^n*ChebyshevU(n-2, -I/2)):
%p seq(simplify(a(n)), n = 0..38); # _Peter Luschny_, Dec 03 2023
%t Join[{1}, Last[#] - First[#] & /@ Partition[Fibonacci[Range[-1, 40]], 4, 1]] (* _Harvey P. Dale_, Sep 30 2011 *)
%t Join[{1}, LinearRecurrence[{1, 1}, {0, 2}, 38]] (* _Jean-François Alcover_, Sep 23 2017 *)
%t Join[{1}, Table[2 Fibonacci[n], {n, 0, 20}]] (* _Eric W. Weisstein_, Oct 03 2017 *)
%t Join[{1}, 2 Fibonacci[Range[0, 20]]] (* _Eric W. Weisstein_, Oct 03 2017 *)
%t CoefficientList[Series[(-1 + x - x^2)/(-1 + x + x^2), {x, 0, 20}], x] (* _Eric W. Weisstein_, Oct 03 2017 *)
%o (PARI) a(n)=if(n,2*fibonacci(n-1),1) \\ _Charles R Greathouse IV_, Mar 14 2012
%o (Haskell)
%o a006355 n = a006355_list !! n
%o a006355_list = 1 : fib2s where
%o fib2s = 0 : map (+ 1) (scanl (+) 1 fib2s)
%o -- _Reinhard Zumkeller_, Mar 20 2013
%o (Magma) [1] cat [Lucas(n) - Fibonacci(n): n in [1..50]]; // _Vincenzo Librandi_, Aug 02 2014
%o (PARI) my(x='x+O('x^50)); Vec((1-x+x^2)/(1-x-x^2)) \\ _Altug Alkan_, Nov 01 2015
%Y Except for initial term, = 2*Fibonacci numbers (A000045).
%Y Essentially the same as A047992, A054886, A055389, A068922, and A090991.
%Y Cf. A097925, A097926. Column 2 in A265584.
%Y Cf. A094570, A118658, A119457.
%K nonn,easy,nice
%O 0,3
%A David M. Bloom
%E Corrected by _T. D. Noe_, Oct 31 2006
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