Search: id:A001998
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%I A001998 M1211 N0468
%S A001998 1,2,4,10,25,70,196,574,1681,5002,14884,44530,133225,399310,1196836,
%T A001998 3589414,10764961,32291602,96864964,290585050,871725625,2615147350,
%U A001998 7845353476,23535971854,70607649841,211822683802,635467254244,1906400965570
%N A001998 Bending a piece of wire of length n+1; walks of length n+1 on a tetrahedron; 
               also non-branched catafusenes with a specified number of condensed 
               hexagons.
%C A001998 The wire stays in the plane, there are n bends, each is R,L or O; turning 
               the wire over does not count as a new figure.
%C A001998 Equivalently, walks of n+1 steps on a tetrahedron, visiting n+2 vertices, 
               with n "corners"; the symmetry group is S4, reversing a walk does 
               not count as different. Simply interpret R,L,O as instructions to 
               turn R, turn L, or retrace the last step. Walks are not self-avoiding.
%C A001998 Also, it appears that a(n) gives the number of equivalence classes of 
               n-tuples of 0, 1, and 2, where two n-tuples are equivalent if one 
               can be obtained from the other by a sequence of operations R and 
               C, where R denotes reversal and C denotes taking the 2's complement 
               (C(x)=2-x). This has been verified up to a(19)=290585050. Example: 
               for n=3 there are ten equivalence classes {000, 222}, {001, 100, 
               122, 221}, {002, 022, 200, 220}, {010, 212}, {011, 110, 112, 211}, 
               {012, 210}, {020, 202}, {021, 102, 120, 201}, {101, 121}, {111}, 
               so a(3)=10. [From John W. Layman (layman(AT)math.vt.edu), Oct 13 
               2009]
%D A001998 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001998 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001998 A. T. Balaban, Enumeration of Cyclic Graphs, pp. 63-105 of A. T. Balaban, 
               ed., Chemical Applications of Graph Theory, Ac. Press, 1976; see 
               p. 75.
%D A001998 A. T. Balaban and F. Harary, Chemical graphs V: enumeration and proposed 
               nomenclature of benzenoid cata-condensed polycyclic aromatic hydrocarbons, 
               Tetrahedron 24 (1968), 2505-2516.
%D A001998 L. W. Beineke and R. E. Pippert, On the enumeration of planar trees of 
               hexagons, Glasgow Math. J., 15 (1974), 131-147.
%D A001998 S. J. Cyvin et al., Unbranched catacondensed polygonal systems containing 
               hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774.
%D A001998 S. J. Cyvin et al., Enumeration of tree-like octagonal systems: catapolyoctagons, 
               ACH Models in Chem. 134 (1997), 55-70.
%D A001998 R. M. Foster, Solution to Problem E185, Amer. Math. Monthly, 44 (1937), 
               50-51.
%D A001998 R. C. Read, The Enumeration of Acyclic Chemical Compounds, pp. 25-61 
               of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. 
               Press, 1976. [I think this reference does not mention this sequence. 
               - N. J. A. Sloane (njas(AT)research.att.com), Aug 10 2006]
%D A001998 Gy. Tasi and F. Mizukami, Quantum algebraic-combinatoric study of the 
               conformational properties of n-alkanes, J. Math. Chemistry, 25, 1999, 
               55-64 (see p. 60).
%H A001998 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 A001998 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 A001998 <a href="Sindx_Fo.html#fold">Index entries for sequences obtained by 
               enumerating foldings</a>
%F A001998 a(n) = if n mod 2 = 0 then ((3^((n-2)/2)+1)/2)^2 else 3^((n-3)/2)+(1/
               4)*(3^(n-2)+1).
%F A001998 G.f.: x*(1-3*x-2*x^2+8*x^3-3*x^4)/((1-x)*(1-3*x)*(1-3*x^2)).
%e A001998 There are 2 ways to bend a piece of wire of length 2 (bend it or not).
%p A001998 A001998 := proc(n) if n = 0 then 1 elif n mod 2 = 1 then (1/4)*(3^n+4*3^((n-1)/
               2)+1) else (1/4)*(3^n+2*3^(n/2)+1); fi; end;
%p A001998 A001998:=(-1+3*z+2*z**2-8*z**3+3*z**4)/(z-1)/(3*z-1)/(3*z**2-1); [Conjectured 
               by S. Plouffe in his 1992 dissertation. Gives sequence with an extra 
               leading 1.]
%Y A001998 Cf. A036359, A002216, A005963, A000228, A001997, A001444, A038766.
%Y A001998 Sequence in context: A027432 A032128 A052829 this_sequence A005817 A148093 
               A148094
%Y A001998 Adjacent sequences: A001995 A001996 A001997 this_sequence A001999 A002000 
               A002001
%K A001998 nonn,nice,easy
%O A001998 0,2
%A A001998 N. J. A. Sloane (njas(AT)research.att.com).
%E A001998 Offset and Maple code corrected by C. L. Mallows, Nov 12, 1999.

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