%I M3288 N1325 #115 Apr 10 2024 04:09:41

%S 1,1,1,1,4,6,19,49,150,442,1424,4522,14924,49536,167367,570285,

%T 1965058,6823410,23884366,84155478,298377508,1063750740,3811803164,

%U 13722384546,49611801980,180072089896,655977266884,2397708652276,8791599732140,32330394085528

%N 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).

%C 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

%C 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

%C 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

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A001683/b001683.txt">Table of n, a(n) for n=2..200</a>

%H Marc J. Beauchamp, <a href="http://d-scholarship.pitt.edu/id/eprint/32702">On Extremal Punctured Spheres</a>, Dissertation, University of Pittsburgh, 2017.

%H F. R. Bernhart & N. J. A. Sloane, <a href="/A001683/a001683.pdf">Correspondence, 1977</a>

%H Douglas Bowman and Alon Regev, <a href="http://arxiv.org/abs/1209.6270">Counting symmetry classes of dissections of a convex regular polygon</a>, arXiv preprint arXiv:1209.6270 [math.CO], 2012. See Th. 29(2).

%H William G. Brown, <a href="http://dx.doi.org/10.1112/plms/s3-14.4.746">Enumeration of Triangulations of the Disk</a>, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.

%H W. G. Brown, <a href="/A002709/a002709.pdf">Enumeration of Triangulations of the Disk</a>, Proc. Lond. Math. Soc. s3-14 (1964) 746-768. [Annotated scanned copy]

%H P. J. Cameron, <a href="http://dx.doi.org/10.1093/qmath/38.2.155">Some treelike objects</a>, 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).

%H P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

%H Malin Christensson, <a href="http://malinc.se/m/ImageTiling.php">Make hyperbolic tilings of images</a>, web page, 2019.

%H O. Devillers, <a href="https://doi.org/10.1016/j.comgeo.2010.10.001">Vertex removal in two-dimensional Delauney triangulation: Speed-up by low degrees optimization</a>, Comp. Geom. 44 (2011) 169.

%H Petr Gregor, Sven Jäger, Torsten Mütze, Joe Sawada, Kaja Wille, <a href="https://arxiv.org/abs/1802.06021">Gray codes and symmetric chains</a>, arXiv:1802.06021 [math.CO], 2018.

%H F. Harary, E. M. Palmer, R. C. Read, <a href="/A000108/a000108_20.pdf">On the cell-growth problem for arbitrary polygons, computer printout, circa 1974</a>

%H F. Harary, E. M. Palmer and R. C. Read, <a href="http://dx.doi.org/10.1016/0012-365X(75)90041-2">On the cell-growth problem for arbitrary polygons</a>, Discr. Math. 11 (1975), 371-389.

%H E. Krasko, A. Omelchenko, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i1p17">Brown's Theorem and its Application for Enumeration of Dissections and Planar Trees</a>, The Electronic Journal of Combinatorics, 22 (2015), #P1.17.

%H C. O. Oakley and R. J. Wisner, <a href="http://www.jstor.org/stable/2310544">Flexagons</a>, The American Mathematical Monthly, Vol. 64, No. 3 (Mar., 1957), pp. 143-154

%H R. C. Read, <a href="/A001004/a001004.pdf">On general dissections of a polygon</a>, Preprint (1974)

%H Hermund A. Torkildsen, <a href="http://www.ieja.net/papers/2008/V4/9-V4-2008.pdf">Counting cluster-tilted algebras of type A_n</a>, International Electronic Journal of Algebra, 4, 2008, 149-158. [From Hermund A. Torkildsen (hermunda(AT)math.ntnu.no), Aug 06 2008]

%H Hermund A. Torkildsen, <a href="http://dx.doi.org/10.1142/S0219498812501332">Colored quivers of type A and the cell-growth problem</a>, J. Algebra and Applications, 12 (2013), #1250133. - From _N. J. A. Sloane_, Jan 22 2013

%F 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.

%F 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

%F a(n) ~ 2^(2*n-4) / (sqrt(Pi) * n^(5/2)). - _Vaclav Kotesovec_, Mar 13 2016

%F 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

%F 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

%p 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);

%t 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 *)

%t 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 *)

%o (PARI) Cat(n)=if(n==floor(n),return(binomial(2*n,n)/(n+1)));0

%o 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

%Y Column k=3 of A295224.

%Y Cf. A007282, A057162.

%Y A row or column of the array in A262586.

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

%K nonn,nice,easy

%O 2,5

%A _N. J. A. Sloane_