Search: id:A005418
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%I A005418 M0771
%S A005418 1,2,3,6,10,20,36,72,136,272,528,1056,2080,4160,8256,16512,32896,65792,
%T A005418 131328,262656,524800,1049600,2098176,4196352,8390656,16781312,33558528,
%U A005418 67117056,134225920,268451840,536887296,1073774592,2147516416,4295032832
%N A005418 Number of n-bead black-white reversible strings; also binary grids; also 
               row sums of Losanitsch's triangle A034851; also number of caterpillar 
               graphs on n nodes.
%C A005418 Equivalently, walks on triangle, visiting n+2 vertices, so length n+1, 
               n "corners"; the symmetry group is S3, reversing a walk does not 
               count as different. Walks are not self-avoiding.
%C A005418 Slavik V. Jablan (jablans(AT)yahoo.com) observes that this is also the 
               number of rational knots and links with n crossings (cf. A018240). 
               See reference.
%C A005418 Number of bit strings of length n, not counting strings which are the 
               end-for-end reversal or the 0-for-1 reversal of each other as different. 
               - Carl Witty (cwitty(AT)newtonlabs.com), Oct 27 2001
%C A005418 The formula given in page 1095 of the Balasubramanian reference can be 
               used to derive this sequence. - Parthasarathy Nambi (PachaNambi(AT)yahoo.com), 
               May 14 2007
%C A005418 a(n) is union A007582(n-1) and A161168(n-1). a(n) is union A007582(n-1) 
               and A063376(n-1). a(n) written in base 2 (see A164370): a(1) = 1, 
               a(2) = 10, a(3) = 11, a(n) for n >= 4: 1010,10100,100100,1001000,
               10001000,100010000,1000010000, ..., i.e. digit 1, (A004526(n-3)) 
               times 0, digit 1, (A004526(n-2)) times 0. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), 
               Aug 14 2009]
%C A005418 Also number of compositions of n up to direction, where a composition 
               is considered equivalent to its reversal. [From Franklin T. Adams-Watters 
               (FrankTAW(AT)Netscape.net), Oct 24 2009]
%D A005418 K. Balasubramanian, "Combinatorial Enumeration of Chemical Isomers", 
               Indian J. Chem., (1978) vol. 16B, pp. 1094-1096. See page 1095.
%D A005418 S. J. Cyvin et al., Theory of polypentagons, J. Chem. Inf. Comput. Sci., 
               33 (1993), 466-474.
%D A005418 N. Hoffman, Binary grids and a related counting problem, 2-Year Coll. 
               Math. J., 9 (1978), 267-272.
%D A005418 Jablan S. and Sazdanovic R., LinKnot: Knot Theory by Computer, World 
               Scientific Press, 2007.
%D A005418 Joseph S. Madachy: Madachy's Mathematical Recreations. New York: Dover 
               Publications, Inc., 1979, p. 46 (first publ. by Charles Scribner's 
               Sons, New York, 1966, under the title: Mathematics on Vacation)
%D A005418 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A005418 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Fxtbook</a>
%H A005418 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A005418 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A005418 S. V. Jablan, <a href="http://www.ams.org./preprints/57/199706/199706-57-001-199706-57-001.html">
               Geometry of Links</a>, XII Yugoslav Geometric Seminar (Novi Sad, 
               1998), Novi Sad J. Math. 29 (1999), no. 3, 121-139.
%H A005418 N. J. A. Sloane, <a href="classic.html#LOSS">Classic Sequences</a>
%H A005418 R. A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Moments of generalized Motzkin paths</a>, J. Integer Sequences, Vol. 
               3 (2000), #00.1.
%H A005418 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               BishopsProblem.html">Link to a section of The World of Mathematics.</
               a>
%H A005418 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               CaterpillarGraph.html">Link to a section of The World of Mathematics.</
               a>
%H A005418 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               BarkerCode.html">Barker Code</a>
%H A005418 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               LosanitschsTriangle.html">Losanitsch's Triangle</a>
%F A005418 2^(n-2) + 2^([n/2]-1).
%F A005418 G.f.: x*(1+2*x)*(1-3*x^2)/((1-4*x^2)*(1-2*x^2)). - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.d\
               e), May 08 2001
%F A005418 a(n) = 6a(n-2)-8a(n-4). a(2n) = A063376(n-1) = 2*a(2n-1); a(2n+1) = A007582(n). 
               - Henry Bottomley (se16(AT)btinternet.com), Jul 14 2001
%F A005418 a(n+2) = 2*a(n+1) - A077957(n) with a(1)=1, a(2)=2 [From Yosu Yurramendi 
               (yosu.yurramendi(AT)ehu.es), Oct 24 2008]
%F A005418 a(n)=2a(n-1)+2a(n-2)-4a(n-3) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), 
               Dec 05 2008]
%p A005418 A005418 := n->2^(n-2)+2^(floor(n/2)-1);
%p A005418 A005418:=-(-1+3*z**2)/(2*z-1)/(2*z**2-1); [Conjectured by S. Plouffe 
               in his 1992 dissertation.]
%Y A005418 Cf. A001998, A001444, A051436, A051437, A007582, A001445.
%Y A005418 Cf. A141447 [From Yosu Yurramendi (yosu.yurramendi(AT)ehu.es), Oct 24 
               2008]
%Y A005418 Sequence in context: A052525 A006606 A120421 this_sequence A002215 A007562 
               A008929
%Y A005418 Adjacent sequences: A005415 A005416 A005417 this_sequence A005419 A005420 
               A005421
%K A005418 nonn,easy,nice
%O A005418 1,2
%A A005418 N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy
%E A005418 Interpretation in terms of walks from Colin Mallows (colinm(AT)research.avayalabs.com).

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