OFFSET
1,3
COMMENTS
Number of 3-Motzkin paths of length n (i.e., lattice paths from (0,0) to (n,0) that do not go below the line y = 0 and consist of steps U = (1,1), D = (1,-1) and three types of steps H = (1,0)) that start with a U step. Example: a(4) = 29 because we have UDUD, UUDD, 9 UDHH paths, 9 UHDH paths and 9 UHHD paths. - Emeric Deutsch, Mar 26 2004
Here, n is the total number of hexagons in the system, which is usually denoted by h in most of the references below. In Cyvin, Brunvoll, and Cyvin (1992), Table 1, p. 28, it seems that the rooted hexagon is "distinguished", and the sequence is shifted by 1. - Petros Hadjicostas, May 26 2019
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
S. J. Cyvin, J. Brunvoll, and B. N. Cyvin, Harary-Read numbers for catafusenes: Complete classification according to symmetry, Journal of Mathematical Chemistry 9(1) (1992), 19-31; see Table 1 (p. 28).
S. J. Cyvin and J. Brunvoll, Generating functions for the Harary-Read numbers classified according to symmetry, Journal of Mathematical Chemistry 9(1) (1992), 33-38.
B. N. Cyvin, E. Brendsdal, J. Brunvoll, and S. J. Cyvin, A class of polygonal systems representing polycyclic conjugated hydrocarbons: Catacondensed monoheptafusenes, Monat. f. Chemie, 125 (1994), 1327-1337; see Eqs (10) and (13) on p. 1330.
S. J. Cyvin, B. N. Cyvin, J. Brunvoll, and E. Brendsdal, Enumeration and Classification of Certain Polygonal Systems Representing Polycyclic Conjugated Hydrocarbons: Annelated Catafusenes, J. Chem. Inform. Comput. Sci., 34 (1994), 1174-1180; see "Two Appendages" in Fig. 1 (p. 1176) for the unsymmetrical case (group C_s).
F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinburgh Math. Soc. 17(2) (1970), 1-13.
Eric Weisstein's World of Mathematics, Fusene.
Wikipedia, Molecular symmetry.
Wikipedia, Point groups in three dimensions.
Wikipedia, Polyhex (mathematics).
Wikipedia, Schoenflies notation.
FORMULA
G.f.: (1/2)*(7*x^2 - 6*x + 1 + (3*x-1)*sqrt(5*x^2-6*x+1))/x^2. - Vladeta Jovovic, Jul 19 2001
a(n) = A002212(n+1) - 3*A002212(n). Convolution of A002212 without the first term with itself. - Emeric Deutsch, Jul 24 2002
a(n) = binomial(2n+2, n+1)/(n+2) + Sum_{k=1..n} binomial(2k, k)*binomial(n-1, k-1)*(3k-2n-3)/((n-k+1)*(k+1)) (n >= 2). - Emeric Deutsch, Mar 26 2004
Recurrence: (n-2)*(n+2)*a(n) = 3*(n-1)*(2*n-1)*a(n-1) - 5*(n-2)*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 08 2012
a(n) ~ 5^(n+1/2)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 08 2012
a(n) = (2/(n+1))*Sum_{m=0..n-1} C(n+1,m)*C(2*n-2*m+2,n-m-1). - Vladimir Kruchinin Oct 18 2022
Let h(n) = hypergeom([-n-2, -n+2], [-n+1/2], -1/4) then a(n) = A002057(n-2)*h(n) = (2*(n-1)/(n+2))*CatalanNumber(n)*h(n). - Peter Luschny, Oct 23 2022
MAPLE
a := n -> binomial(2*n+2, n+1)/(n+2) + add(binomial(2*k, k)*binomial(n-1, k-1)*(3*k-2*n-3)/(n-k+1)/(k+1), k=1..n): 0, seq(a(n), n=2..23);
# Alternative:
a := n -> (2*(n - 1)/(n + 2))*(binomial(2*n, n) / (n + 1))*hypergeom([-n-2, -n+2], [-n + 1/2], -1/4): seq(simplify(a(n)), n = 1..26); # Peter Luschny, Oct 23 2022
MATHEMATICA
a[n_] = Binomial[2n+2, n+1]/(n+2) + Sum[Binomial[2k, k]*Binomial[n-1, k-1]*(3k-2n-3)/(n-k+1)/(k+1), {k, 1, n}];
a /@ Range[23] (* Jean-François Alcover, Jul 13 2011, after Maple *)
Table[SeriesCoefficient[(1/2)*(7*x^2-6*x+1+(3*x-1)*Sqrt[5*x^2-6*x+1])/x^2, {x, 0, n}], {n, 1, 23}] (* Vaclav Kotesovec, Oct 08 2012 *)
PROG
(PARI) x='x+O('x^66); concat([0], Vec((1/2)*(7*x^2-6*x+1+(3*x-1)*sqrt(5*x^2-6*x+1))/x^2)) \\ Joerg Arndt, May 04 2013
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, Jul 19 2001
STATUS
approved