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A005418
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Number of (n-1) bead black-white reversible strings; also binary grids; also row sums of Losanitsch's triangle A034851; also number of caterpillar graphs on n nodes.
(Formerly M0771)
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56
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1, 2, 3, 6, 10, 20, 36, 72, 136, 272, 528, 1056, 2080, 4160, 8256, 16512, 32896, 65792, 131328, 262656, 524800, 1049600, 2098176, 4196352, 8390656, 16781312, 33558528, 67117056, 134225920, 268451840, 536887296, 1073774592, 2147516416, 4295032832
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OFFSET
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1,2
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COMMENTS
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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. - Colin Mallows.
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.
Number of bit strings of length (n-1), 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
The formula given in page 1095 of the Balasubramanian reference can be used to derive this sequence. - Parthasarathy Nambi, May 14 2007
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. - Jaroslav Krizek, Aug 14 2009
Also number of compositions of n up to direction, where a composition is considered equivalent to its reversal, see example. - Franklin T. Adams-Watters, Oct 24 2009
Number of normally non-isomorphic realizations of the associahedron of type I starting with dimension 2 in Ceballos et al. - Tom Copeland, Oct 19 2011
Number of fibonacenes with n+2 hexagons. See the Balaban and the Dobrynin references. - Emeric Deutsch, Apr 21 2013
From the point of view of binary grids, it is a (1,n)-rectangular grid. A225826 to A225834 are the numbers of binary pattern classes in the (m,n)-rectangular grid, 1 < m < 11 . - Yosu Yurramendi, May 19 2013
Number of n-vertex difference graphs (bipartite 2K_2-free graphs) [Peled&Sun, Thm. 9]. - Falk Hüffner, Jan 10 2016
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REFERENCES
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A. T. Balaban, Chemical graphs. Part 50. Symmetry and enumeration of fibonacenes (unbranched catacondensed benzenoids isoarithmic with helicenes and zigzag catafusenes, MATCH: Commun. Math. Comput. Chem., 24 (1989) 29-38.
K. Balasubramanian, "Combinatorial Enumeration of Chemical Isomers", Indian J. Chem., (1978) vol. 16B, pp. 1094-1096. See page 1095.
Dymacek, Wayne M. Steinhaus graphs. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 399--412, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561065 (81f:05120) - N. J. A. Sloane, May 30 2012
Jablan S. and Sazdanovic R., LinKnot: Knot Theory by Computer, World Scientific Press, 2007.
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)
C. A. Pickover, Keys to Infinity, Wiley 1995, p. 75.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Joerg Arndt, Matters Computational (The Fxtbook), p.151, p. 733
C. Ceballos, F. Santos, and G. Ziegler, Many Non-equivalent Realizations of the Associahedron, p. 15 and 26, arXiv:1109.5544 [math.MG], 2011-2013.
S. J. Cyvin et al., Theory of polypentagons, J. Chem. Inf. Comput. Sci., 33 (1993), 466-474.
A. A. Dobrynin, On the Wiener index of fibonacenes, MATCH: Commun. Math. Comput. Chem, 64 (2010) 707-726.
T. A. Gittings, Minimum braids: a complete invariant of knots and links, arXiv:math/0401051 [math.GT], 2004. - N. J. A. Sloane, Jan 18 2013
R. K. Guy, Letter to N. J. A. Sloane, Nov 1978
N. Hoffman, Binary grids and a related counting problem, 2-Year Coll. Math. J., 9 (1978), 267-272.
S. V. Jablan, Geometry of Links, XII Yugoslav Geometric Seminar (Novi Sad, 1998), Novi Sad J. Math. 29 (1999), no. 3, 121-139.
U. N. Peled and F. Sun, Enumeration of difference graphs, Discrete Appl. Math., 60 (1995), 311-318.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.
Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.
A. Regev, Remarks on two-eared triangulations, arXiv preprint arXiv:1309.0743 [math.CO], 2013-2014.
N. J. A. Sloane, Classic Sequences
R. A. Sulanke, Moments of generalized Motzkin paths, J. Integer Sequences, Vol. 3 (2000), #00.1.
Eric Weisstein's World of Mathematics, Barker Code
Eric Weisstein's World of Mathematics, Bishops Problem
Eric Weisstein's World of Mathematics, Caterpillar Graph
Eric Weisstein's World of Mathematics, Losanitsch's Triangle
A. Yajima, How to calculate the number of stereoisomers of inositol-homologs, Bull. Chem. Soc. Jpn. 2014, 87, 1260-1264 | doi:10.1246/bcsj.20140204. See Tables 1 and 2 (and text). - N. J. A. Sloane, Mar 26 2015
Index entries for linear recurrences with constant coefficients, signature (2,2,-4)
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FORMULA
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a(n) = 2^(n-2) + 2^([n/2]-1).
G.f.: -x*(-1+3*x^2) / ( (2*x-1)*(2*x^2-1) ). - Simon Plouffe in his 1992 dissertation
G.f.: x*(1+2*x)*(1-3*x^2)/((1-4*x^2)*(1-2*x^2)), not reduced. - Wolfdieter Lang, May 08 2001
a(n) = 6*a(n-2)-8*a(n-4). a(2*n) = A063376(n-1) = 2*a(2n-1); a(2*n+1) = A007582(n). - Henry Bottomley, Jul 14 2001
a(n+2) = 2*a(n+1) - A077957(n) with a(1) = 1, a(2) = 2. - Yosu Yurramendi, Oct 24 2008
a(n) = 2*a(n-1)+2*a(n-2)-4*a(n-3) - Jaume Oliver Lafont, Dec 05 2008
G.f.: G(0) ; G(k) = 1 + 2*x/(1 - x*(1+2^(k+1))/(x*(1+2^(k+1)) + (1+2^k)/G(k+1))) ; (continued fraction). - Sergei N. Gladkovskii, Dec 12 2011
a(2*n) = 2*a(2*n-1) and a(2*n+1) = a(2*n) + 4^(n-1) with a(1) = 1. - Johannes W. Meijer, Aug 26 2013
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EXAMPLE
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a(5)=10 because there are 16 compositions of 5 (shown as <vectors>) but only 10 equivalence classes (shown as {sets}): {<5>}, {<4,1>,<1,4>}, {<3,2>,<2,3>}, {<3,1,1>,<1,1,3>}, {<1,3,1>},{<2,2,1>,<1,2,2>}, {<2,1,2>}, {<2,1,1,1>,<1,1,1,2>}, {<1,2,1,1>,<1,1,2,1>}, {<1,1,1,1,1>}. - Geoffrey Critzer, Nov 02 2012
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MAPLE
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A005418 := n->2^(n-2)+2^(floor(n/2)-1): seq(A005418(n), n=1..34);
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MATHEMATICA
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LinearRecurrence[{2, 2, -4}, {1, 2, 3}, 40] (* or *) Table[2^(n-2)+2^(Floor[n/2]-1), {n, 40}] (* Harvey P. Dale, Jan 18 2012 *)
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PROG
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(Haskell)
a005418 n = sum $ a034851_row (n - 1) -- Reinhard Zumkeller, Jan 14 2012
(PARI) A005418(n)= 2^(n-2) + 2^(n\2-1); \\ Joerg Arndt, Sep 16 2013
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CROSSREFS
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Column 2 of A277504.
Cf. A001998, A001444, A051436, A051437, A007582, A001445, A032085.
Sequence in context: A052525 A006606 A120421 * A002215 A007562 A222855
Adjacent sequences: A005415 A005416 A005417 * A005419 A005420 A005421
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane, R. K. Guy
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STATUS
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approved
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