%I A001683 M3288 N1325
%S A001683 1,1,1,1,4,6,19,49,150,442,1424,4522,14924,49536,167367,570285,
%T A001683 1965058,6823410,23884366,84155478,298377508,1063750740,3811803164,
%U A001683 13722384546,49611801980,180072089896,655977266884,2397708652276
%N 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).
%C A001683 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. [From Hermund A. Torkildsen (hermunda(AT)math.ntnu.no), Aug 
               06 2008]
%D A001683 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001683 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001683 W. G. Brown, Enumeration of triangulations of the disk, Proc. London 
               Math. Soc., 14 (1964), 746-768.
%D A001683 F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for 
               arbitrary polygons, Discr. Math. 11 (1975), 371-389.
%D A001683 C. O. Oakley and R. J. Wisner, Flexagons, Amer. Math. Monthly, 64 (1957), 
               143-154.
%D A001683 Torkildsen, Hermund A., 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]
%H A001683 T. D. Noe, <a href="b001683.txt">Table of n, a(n) for n=2..200</a>
%H A001683 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Sequences realized by oligomorphic permutation groups</a>, J. Integ. 
               Seqs. Vol. 3 (2000), #00.1.5.
%H A001683 Torkildsen, Hermund A., <a href="http://www.ieja.net/papers/2008/V4/9-V4-2008.pdf">
               Counting cluster-tilted algebras of type A_n</a> [From Hermund A. 
               Torkildsen (hermunda(AT)math.ntnu.no), Aug 06 2008]
%F A001683 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 A001683 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 (callan(AT)stat.wisc.edu), Aug 01 2004
%p A001683 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 A001683 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, 1, 20}] - Robert A. Russell (russell(AT)post.harvard.edu), Dec 
               11 2004
%Y A001683 Cf. A007282, A057162.
%Y A001683 Sequence in context: A064035 A010364 A110391 this_sequence A053892 A013126 
               A012969
%Y A001683 Adjacent sequences: A001680 A001681 A001682 this_sequence A001684 A001685 
               A001686
%K A001683 nonn,nice,easy
%O A001683 2,5
%A A001683 N. J. A. Sloane (njas(AT)research.att.com).