

A001683


Number of onesided triangulations of the disk; or flexagons of order n; or unlabeled plane trivalent trees (n2 internal vertices, all of degree 3 and hence n leaves).
(Formerly M3288 N1325)


25



1, 1, 1, 1, 4, 6, 19, 49, 150, 442, 1424, 4522, 14924, 49536, 167367, 570285, 1965058, 6823410, 23884366, 84155478, 298377508, 1063750740, 3811803164, 13722384546, 49611801980, 180072089896, 655977266884, 2397708652276, 8791599732140, 32330394085528
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OFFSET

2,5


COMMENTS

a(n) is the number of triangulations of an ngon (equivalently, the number of vertices of the (n  3)dimensional associahedron) modulo the cyclic action [Bowman and Negev].  N. J. A. Sloane, Dec 29 2012
a(n) is also the number of nonisomorphic clustertilted algebras of type A_(n3), for n greater than or equal to 5. Equivalently it is the number of nonisomorphic quivers in the mutation class of any quiver with underlying graph A_(n3) for n greater than or equal to 5.  Hermund A. Torkildsen (hermunda(AT)math.ntnu.no), Aug 06 2008


REFERENCES

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

T. D. Noe, Table of n, a(n) for n=2..200
Marc J. Beauchamp, On Extremal Punctured Spheres, Dissertation, University of Pittsburgh, 2017.
F. R. Bernhart & N. J. A. Sloane, Correspondence, 1977
Douglas Bowman and Alon Regev, Counting symmetry classes of dissections of a convex regular polygon, arXiv preprint arXiv:1209.6270 [math.CO], 2012. See Th. 29(2).
William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s314 (1964) 746768.
W. G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s314 (1964) 746768. [Annotated scanned copy]
P. J. Cameron, Some treelike objects, Quart. J. Math. Oxford, 38 (1987), 155183. See p. 163, line 4, but note that the formula given there has many typos (see the correct version given here).
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
O. Devillers, Vertex removal in twodimensional Delauney triangulation: Speedup by low degrees optimization, Comp. Geom. 44 (2011) 169.
Petr Gregor, Sven Jäger, Torsten Mütze, Joe Sawada, Kaja Wille, Gray codes and symmetric chains, arXiv:1802.06021 [math.CO], 2018.
F. Harary, E. M. Palmer, R. C. Read, On the cellgrowth problem for arbitrary polygons, computer printout, circa 1974
F. Harary, E. M. Palmer and R. C. Read, On the cellgrowth problem for arbitrary polygons, Discr. Math. 11 (1975), 371389.
E. Krasko, A. Omelchenko, Brown's Theorem and its Application for Enumeration of Dissections and Planar Trees, The Electronic Journal of Combinatorics, 22 (2015), #P1.17.
C. O. Oakley and R. J. Wisner, Flexagons, The American Mathematical Monthly, Vol. 64, No. 3 (Mar., 1957), pp. 143154
R. C. Read, On general dissections of a polygon, Preprint (1974)
Hermund A. Torkildsen, Counting clustertilted algebras of type A_n, International Electronic Journal of Algebra, 4, 2008, 149158. [From Hermund A. Torkildsen (hermunda(AT)math.ntnu.no), Aug 06 2008]
Hermund A. Torkildsen, Colored quivers of type A and the cellgrowth problem, J. Algebra and Applications, 12 (2013), #1250133.  From N. J. A. Sloane, Jan 22 2013


FORMULA

a(n) = C(n2)/n + C(n/21)/2 + (2/3)*C(n/31), where C(n) = Catalan(n) (A000108) and terms are omitted if their subscripts are not integers.
G.f.: (6 + (1  4*x)^(3/2) + 6*x  3*(1  4*x^2)^(1/2)  4*(1  4*x^3)^(1/2))/12.  David Callan, Aug 01 2004
a(n) ~ 2^(2*n4) / (sqrt(Pi) * n^(5/2)).  Vaclav Kotesovec, Mar 13 2016


MAPLE

C := n>binomial(2*n, n)/(n+1); c := x>if whattype(x) = integer then C(x) else 0; fi; A001683 := n>C(n2)/n + c(n/21)/2+(2/3)*c(n/31);


MATHEMATICA

p=3; Table[Binomial[(p1)n, n]/(((p2)n+1)((p2)n+2)) +If[OddQ[n], 0, Binomial[(p1)n/2, n/2]/((p2)n+2)]+Plus @@ Map[EulerPhi[ # ]Binomial[((p1)n+1)/#, (n1)/# ]/((p1)n+1)&, Complement[Divisors[GCD[p, n1]], {1}]], {n, 1, 20}] (* Robert A. Russell, Dec 11 2004 *)
Rest[Rest[CoefficientList[Series[(6 + (1  4 x)^(3/2) + 6 x  3(1  4 x^2)^(1/2)  4 (1  4 x^3)^(1/2))/12, {x, 0, 33}], x]]] (* Vincenzo Librandi, Nov 25 2015 *)


PROG

(PARI) Cat(n)=if(n==floor(n), return(binomial(2*n, n)/(n+1))); 0
for(n=2, 100, print1(Cat(n2)/n+Cat(n/21)/2+(2/3)*Cat(n/31), ", ")) \\ Derek Orr, Feb 26 2017


CROSSREFS

Column k=3 of A295224.
Cf. A007282, A057162.
A row or column of the array in A262586.
Sequence in context: A010364 A110391 A197460 * A053892 A013126 A012969
Adjacent sequences: A001680 A001681 A001682 * A001684 A001685 A001686


KEYWORD

nonn,nice,easy


AUTHOR

N. J. A. Sloane


STATUS

approved



