A001683 Number of one-sided triangulations of the disk; or flexagons of order n; or unlabeled plane trivalent trees (n-2 internal vertices, all of degree 3 and hence n leaves).

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

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

a(n) is the number of triangulations of an n-gon (equivalently, the number of vertices of the (n - 3)-dimensional associahedron) modulo the cyclic action [Bowman and Regev]. - N. J. A. Sloane, Dec 29 2012

a(n) is also the number of non-isomorphic cluster-tilted algebras of type A_(n-3), for n greater than or equal to 5. Equivalently it is the number of non-isomorphic quivers in the mutation class of any quiver with underlying graph A_(n-3) for n greater than or equal to 5. - Hermund A. Torkildsen (hermunda(AT)math.ntnu.no), Aug 06 2008

Number of oriented polyominoes composed of n-2 triangular cells of the hyperbolic regular tiling with Schläfli symbol {3,oo}. A stereographic projection of this tiling on the Poincaré disk can be obtained via the Christensson link. For oriented polyominoes, chiral pairs are counted as two. - Robert A. Russell, Jan 20 2024

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. s3-14 (1964) 746-768.
• W. G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768. [Annotated scanned copy]
• P. J. Cameron, Some treelike objects, Quart. J. Math. Oxford, 38 (1987), 155-183. 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.
• Malin Christensson, Make hyperbolic tilings of images, web page, 2019.
• O. Devillers, Vertex removal in two-dimensional Delauney triangulation: Speed-up 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 cell-growth problem for arbitrary polygons, computer printout, circa 1974
• F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389.
• 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. 143-154
• R. C. Read, On general dissections of a polygon, Preprint (1974)
• Hermund A. Torkildsen, Counting cluster-tilted algebras of type A_n, International Electronic Journal of Algebra, 4, 2008, 149-158. [From Hermund A. Torkildsen (hermunda(AT)math.ntnu.no), Aug 06 2008]
• Hermund A. Torkildsen, Colored quivers of type A and the cell-growth problem, J. Algebra and Applications, 12 (2013), #1250133. - From _N. J. A. Sloane_, Jan 22 2013

Crossrefs

Column k=3 of A295224.

Cf. A007282, A057162.

A row or column of the array in A262586.

Polyominoes: A000207 (unoriented), A369314 (chiral), A208355(n-1) (achiral), A005034 {4,oo}, A007173 {3,3,oo}.

Programs

Maple

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

Mathematica

p=3; Table[Binomial[(p-1)n, n]/(((p-2)n+1)((p-2)n+2)) +If[OddQ[n], 0, Binomial[(p-1)n/2, n/2]/((p-2)n+2)]+Plus @@ Map[EulerPhi[ # ]Binomial[((p-1)n+1)/#, (n-1)/# ]/((p-1)n+1)&, Complement[Divisors[GCD[p, n-1]], {1}]], {n, 0, 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 *)

PARI

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

Formula

a(n) = C(n-2)/n + C(n/2-1)/2 + (2/3)*C(n/3-1), 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*n-4) / (sqrt(Pi) * n^(5/2)). - Vaclav Kotesovec, Mar 13 2016

a(n+2) = A000207(n) + A369314(n) = 2*A000207(n) - A208355(n-1) = 2*A369314(n) + A208355(n-1). - Robert A. Russell, Jan 19 2024

G.f.: z^2 * (4*G(z) - G(z)^2 + 3*G(z^2) + 4*z*G(z^3)) / 6, where G(z) = 1 + z*G(z)^2 is the g.f. for A000108. - Robert A. Russell, Apr 06 2024