The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A001998 Bending a piece of wire of length n+1; walks of length n+1 on a tetrahedron; also non-branched catafusenes with n+2 condensed hexagons. (Formerly M1211 N0468) 21
 1, 2, 4, 10, 25, 70, 196, 574, 1681, 5002, 14884, 44530, 133225, 399310, 1196836, 3589414, 10764961, 32291602, 96864964, 290585050, 871725625, 2615147350, 7845353476, 23535971854, 70607649841, 211822683802, 635467254244, 1906400965570, 5719200505225, 17157599124190 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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. 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. 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. - John W. Layman, Oct 13 2009 There exists a bijection between chains of n+2 hexagons and the above described equivalence classes of n-tuples of 0,1, and 2. Namely, for a given chain of n+2 hexagons we take the sequence of the numbers of vertices of degree 2 (0, 1, or 2) between the consecutive contact vertices on one side of the chain; switching to the other side we obtain the 2's complement of this sequence; reversing the order of the hexagons, we obtain the reverse sequence. The inverse mapping is straightforward. For example, to a linear chain of 7 hexagons there corresponds the 5-tuple 11111. - Emeric Deutsch, Apr 22 2013 If we treat two wire bends (or walks, or tuples) related by turning over (or reversing) as different in any of the above-given interpretations of this sequence, we get A007051 (or A124302). Also, a(n-1) is the sum of first 3 terms in n-th row of A284949, see crossrefs therein. - Andrey Zabolotskiy, Sep 29 2017 a(n-1) is the number of color patterns (set partitions) in an unoriented row of length n using 3 or fewer colors (subsets). - Robert A. Russell, Oct 28 2018 REFERENCES 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. S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Enumeration of tree-like octagonal systems: catapolyoctagons, ACH Models in Chem. 134 (1997), 55-70. 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, Aug 10 2006] N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Indranil Ghosh, Table of n, a(n) for n = 0..2092 (terms 0..500 from T. D. Noe) A. T. Balaban, J. Brunvoll, B. N. Cyvin, and S. J. Cyvin, Enumeration of branched catacondensed benzenoid hydrocarbons and their numbers of Kekulé structures, Tetrahedron 44(1) (1988), 221-228. See Eq. (6), p. 223. 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. Christian Barrientos and Sarah Minion, On the Graceful Cartesian Product of Alpha-Trees, Theory and Applications of Graphs, Vol. 4: Iss. 1, Article 3, 2017. (It mentions this sequence on p. 7.) L. W. Beineke and R. E. Pippert, On the enumeration of planar trees of hexagons, Glasgow Math. J., 15 (1974), 131-147. L. W. Beineke and R. E. Pippert, On the enumeration of planar trees of hexagons, Glasgow Math. J., 15 (1974), 131-147 [annotated scanned copy]. S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Isomer enumeration of some polygonal systems representing polycyclic conjugated hydrocarbons, Journal of Molecular structure 376 (Issues 1-3) (1996), 495-505. See Table 2 on p. 501. S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Unbranched catacondensed polygonal systems containing hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774. R. M. Foster, Solution to Problem E185, Amer. Math. Monthly, 44 (1937), 50-51. R. M. Foster, Solution to Problem E185, Amer. Math. Monthly, 44 (1937), 50-51 [annotated scanned copy]. F. Harary and R. W. Robinson, Tapeworms, Unpublished manuscript, circa 1973. (Annotated scanned copy) Thomas M. Liggett, Wenpin Tang, One-dependent hard-core processes and colorings of the star graph, arXiv:1804.06877 [math.PR], 2018. Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992 Gyula Tasi and Fujio Mizukami, Quantum algebraic-combinatoric study of the conformational properties of n-alkanes, J. Math. Chemistry, 25, 1999, 55-64 (see p. 60). Index entries for linear recurrences with constant coefficients, signature (4,0,-12,9). FORMULA 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). G.f.: (1-2*x-4*x^2+6*x^3) / ((1-x)*(1-3*x)*(1-3*x^2)). - Corrected by Colin Barker, May 15 2016 a(n) = 4*a(n-1)-12*a(n-3)+9*a(n-4), with a(0)=1, a(1)=2, a(2)=4, a(3)=10. - Harvey P. Dale, Apr 10 2013 a(n) = (1+3^n+3^(1/2*(-1+n))*(2-2*(-1)^n+sqrt(3)+(-1)^n*sqrt(3)))/4. - Colin Barker, May 15 2016 E.g.f.: (2*sqrt(3)*sinh(sqrt(3)*x) + 3*exp(2*x)*cosh(x) + 3*cosh(sqrt(3)*x))/6. - Ilya Gutkovskiy, May 15 2016 From Robert A. Russell, Oct 28 2018: (Start) a(n-1) = (A124302(n) + A182522(n)) / 2 = A124302(n) - A107767(n-1) = A107767(n-1) + A182522(n). a(n-1) = Sum_{j=1..k} (S2(n,j) + Ach(n,j)) / 2, where k=3 is the maximum number of colors, S2 is the Stirling subset number A008277, and Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)). a(n-1) = A057427(n) + A056326(n) + A056327(n). (End) EXAMPLE There are 2 ways to bend a piece of wire of length 2 (bend it or not). For n=4 and a(n-1)=10, the 6 achiral patterns are AAAA, AABB, ABAB, ABBA, ABCA, and ABBC.  The 4 chiral pairs are AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB. - Robert A. Russell, Oct 28 2018 MAPLE 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; A001998:=(-1+3*z+2*z**2-8*z**3+3*z**4)/(z-1)/(3*z-1)/(3*z**2-1); # conjectured by Simon Plouffe in his 1992 dissertation; gives sequence with an extra leading 1 MATHEMATICA a[n_?OddQ] := (1/4)*(3^n + 4*3^((n - 1)/2) + 1); a[n_?EvenQ] := (1/4)*(3^n + 2*3^(n/2) + 1); Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Jan 25 2013, from formula *) LinearRecurrence[{4, 0, -12, 9}, {1, 2, 4, 10}, 30] (* Harvey P. Dale, Apr 10 2013 *) Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2, k] + Ach[n-2, k-1] + Ach[n-2, k-2]] (* A304972 *) k=3; Table[Sum[StirlingS2[n, j]+Ach[n, j], {j, k}]/2, {n, 40}] (* Robert A. Russell, Oct 28 2018 *) PROG (PARI) Vec((1-2*x-4*x^2+6*x^3)/((1-x)*(1-3*x)*(1-3*x^2)) + O(x^50)) \\ Colin Barker, May 15 2016 (GAP) a:=[];; for n in [2..45] do if n mod 2 =0 then Add(a, ((3^((n-2)/2)+1)/2)^2); else Add(a,  3^((n-3)/2)+(1/4)*(3^(n-2)+1)); fi; od; a; # Muniru A Asiru, Oct 28 2018 CROSSREFS Cf. A036359, A002216, A005963, A000228, A001997, A001444, A038766. Column 3 of A320750, offset by one. Column k = 0 of A323942, offset by two. Cf. A124302 (oriented), A107767 (chiral), A182522 (achiral), with varying offsets. Sequence in context: A032128 A052829 A339295 * A005817 A302093 A292617 Adjacent sequences:  A001995 A001996 A001997 * A001999 A002000 A002001 KEYWORD nonn,nice,easy AUTHOR EXTENSIONS Offset and Maple code corrected by Colin Mallows, Nov 12 1999 Term added by Robert A. Russell, Oct 30 2018. STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified June 19 12:50 EDT 2021. Contains 345129 sequences. (Running on oeis4.)