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). (Formerly M3288 N1325)

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

2,5

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

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

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 *)

Prog

(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

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}.

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