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%I M1048 #168 Oct 23 2023 21:43:00
%S 1,1,1,1,2,4,7,11,17,27,44,72,117,189,305,493,798,1292,2091,3383,5473,
%T 8855,14328,23184,37513,60697,98209,158905,257114,416020,673135,
%U 1089155,1762289,2851443,4613732,7465176,12078909,19544085,31622993
%N a(n) = Sum_{k=0..floor(n/4)} binomial(n-2k,2k).
%C The Twopins/t sequence (see Guy).
%C Number of closed walks of length n at a vertex of the graph with adjacency matrix [1,1,0,0;0,0,0,1;1,0,0,0;0,0,1,1]. - _Paul Barry_, Mar 15 2004
%C a(n+3) = number of n-bit sequences that avoid both 010 and 0110. Example: for n=3, there are 8 3-bit sequences and only 010 fails to qualify, so a(6)=7. - _David Callan_, Mar 25 2004
%C a(n) is the number of length n binary words that have an even number of 0's and every 0 is immediately followed by a 1. a(6) = 7 because we have: 010111, 011011, 011101, 101011, 101101, 110101, 111111. - _Geoffrey Critzer_, Jan 08 2014
%C a(n) = number of vertices of the Fibonacci cube Gamma(n-1) having an even number of ones. The Fibonacci cube Gamma(n) can be defined as the graph whose vertices are the binary strings of length n without two consecutive 1's and in which two vertices are adjacent when their Hamming distance is exactly 1. Example: a(4) = 2; indeed, the Fibonacci cube Gamma(3) has the five vertices 000, 010, 001, 100, 101, two of which have an even number of ones. See the E. Munarini et al. reference, p. 323. - _Emeric Deutsch_, Jun 28 2015
%C a(n) is the number of even permutations p of 1,2,...,n such that |p(i)-i| <= 1 for i=1,2,...,n. - _Dmitry Efimov_, Jan 08 2016
%C This sequence (prefixed with 0) is an autosequence of the first kind, whose second kind companion is (2 followed by abs(A111734)). - _Jean-François Alcover_, Oct 30 2017
%D John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, p. 205 of first edition.
%D R. K. Guy, Anyone for Twopins?, in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15.
%D David J. C. MacKay, Information Theory, Inference and Learning Algorithms, CUP, 2003, p. 251.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D E. L. Tan, On the cycle graph of a graph and inverse cycle graphs, Ph.D. Dissertation, Univ. of Philippines, Diliman, Quezon City, 1987.
%D E. L. Tan, On Fibonacci numbers and cycle graphs, Matimyas Matemaka (Published by the Mathematical Society of the Philippines), 13 (No. 2, 1990), 1-4.
%H T. D. Noe, <a href="/A005252/b005252.txt">Table of n, a(n) for n = 0..500</a>
%H R. Austin and R. K. Guy, <a href="http://www.fq.math.ca/Scanned/16-1/austin.pdf">Binary sequences without isolated ones</a>, Fib. Quart., 16 (1978), 84-86.
%H R. K. Guy, <a href="/A005251/a005251_1.pdf">Anyone for Twopins?</a>, in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15. [Annotated scanned copy, with permission]
%H V. C. Harris, C. C. Styles, <a href="http://www.fq.math.ca/Scanned/2-4/harris.pdf">A generalization of Fibonacci numbers</a>, Fib. Quart. 2 (1964) 277-289, sequence u(n,2,2).
%H V. E. Hoggatt, Jr., <a href="/A005676/a005676.pdf">7-page typed letter to N. J. A. Sloane with suggestions for new sequences</a>, circa 1977.
%H V. E. Hoggatt, Jr. and M. Bicknell, <a href="http://www.fq.math.ca/Scanned/7-4/hoggatt-a.pdf">Diagonal sums of generalized Pascal triangles</a>, Fib. Quart., 7 (1969), 341-358, 393.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=424">Encyclopedia of Combinatorial Structures 424</a>
%H Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Janjic/janjic73.html">Binomial Coefficients and Enumeration of Restricted Words</a>, Journal of Integer Sequences, 2016, Vol 19, #16.7.3
%H S. Klavzar, <a href="http://www.fmf.uni-lj.si/~klavzar/preprints/FibonacciCubesRevised.pdf">Structure of Fibonacci cubes: a survey</a>, J. Comb. Optim. 25 (2013), 505-522, DOI:<a href="https://doi.org/10.1007/s10878-011-9433-z">10.1007/s10878-011-9433-z</a>.
%H E. Munarini, N. Z. Salvi, <a href="https://doi.org/10.1016/S0012-365X(01)00407-1">Structural and enumerative properties of the Fibonacci cubes</a>, Discrete Math., 255, 2002, 317-324.
%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 E. L. Tan, <a href="/A005252/a005252.pdf">Letter to N. J. A. Sloane, Feb 1992</a>
%H OEIS Wiki, <a href="https://oeis.org/wiki/Autosequence">Autosequence</a>.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,1).
%F Second differences give sequence shifted twice. - E. L. Tan, Univ. Phillipines.
%F G.f.: (1-x)/((1-x-x^2)*(1-x+x^2)). _Simon Plouffe_ in his 1992 dissertation.
%F From _Paul Barry_, Mar 15 2004: (Start)
%F a(n) = Fibonacci(n+1)/2 + A010892(n)/2;
%F a(n) = (((1+sqrt(5))/2)^(n+1)/sqrt(5) - ((1-sqrt(5))/2)^(n+1)/sqrt(5) + cos(Pi*n/3) + sin(Pi*n/3)/sqrt(3))/2. (End)
%F a(n) = 2*a(n-1) - a(n-2) + a(n-4); a(0) = a(1) = a(2) = a(3) = 1. - _Philippe Deléham_, May 01 2006
%F a(n) = A173021(2^(n-1) - 1) for n > 0. - _Reinhard Zumkeller_, Feb 07 2010
%F Limit_{n->oo} a(n)/a(n+1) = (sqrt(5) - 1)/2. - _Sergei N. Gladkovskii_, Jan 05 2014
%F G.f.: (1 + Q(0)*x^4/2)/(1-x), where Q(k) = 1 + 1/(1 - x*( 4*k + 2 - x + x^3)/( x*( 4*k + 4 - x + x^3) + 1/Q(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jan 07 2014
%F a(n) = Fibonacci(n+1) + (-1)^(n+1)*A106511(n+2). - _Katharine Ahrens_, May 05 2019
%F E.g.f.: exp(x/2)*(15*(cos(sqrt(3)*x/2) + cosh(sqrt(5)*x/2)) + 5*sqrt(3)*sin(sqrt(3)*x/2) + 3*sqrt(5)*sinh(sqrt(5)*x/2))/30. - _Stefano Spezia_, Aug 03 2022
%p ZL:=[S, {a = Atom, b = Atom, S = Prod(X,Sequence(Prod(X,X))), X = Sequence(b,card >= 2)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=2..40); # _Zerinvary Lajos_, Mar 26 2008
%t Table[Sum[Binomial[n-2k,2k],{k,0,Floor[n/4]}],{n,0,50}] (* or *) LinearRecurrence[{2,-1,0,1},{1,1,1,1},50] (* _Harvey P. Dale_, Dec 09 2011 *)
%t Table[HypergeometricPFQ[{1/4-n/4, 1/2-n/4, 3/4-n/4, -n/4}, {1/2, 1/2-n/2, -n/2}, 16], {n, 0, 38}] (* _Jean-François Alcover_, Oct 04 2012 *)
%o (Haskell)
%o a005252 n = sum $ map (\x -> a007318 (n - x) x) [0, 2 .. 2 * div n 4]
%o -- _Reinhard Zumkeller_, Jul 05 2013
%o (PARI) Vec((1-x)/((1-x-x^2)*(1-x+x^2)) + O(x^100)) \\ _Altug Alkan_, Jan 08 2015
%o (PARI) a(n) = fibonacci(n+1)>>1 + (n%6<2); \\ _Kevin Ryde_, Apr 29 2021
%o (Magma) I:=[1,1,1,1]; [n le 4 select I[n] else 2*Self(n-1)-Self(n-2)+Self(n-4): n in [1..50]]; // _Vincenzo Librandi_, Jan 09 2016
%Y First differences of A024490.
%Y Cf. A005251, A005676, A007318, A010892.
%Y Cf. A106511, A111734, A173021.
%K nonn,easy,nice
%O 0,5
%A _N. J. A. Sloane_
%E More terms from (and formula corrected by) _James A. Sellers_, Feb 06 2000
%E Definition revised at the suggestion of Alessandro Orlandi by _N. J. A. Sloane_, Aug 16 2009
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