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A001683
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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)
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12
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 2,5
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COMMENTS
| 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]
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REFERENCES
| W. G. Brown, Enumeration of triangulations of the disk, Proc. London Math. Soc., 14 (1964), 746-768.
F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389.
C. O. Oakley and R. J. Wisner, Flexagons, Amer. Math. Monthly, 64 (1957), 143-154.
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).
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]
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LINKS
| T. D. Noe, Table of n, a(n) for n=2..200
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Torkildsen, Hermund A., Counting cluster-tilted algebras of type A_n [From Hermund A. Torkildsen (hermunda(AT)math.ntnu.no), Aug 06 2008]
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FORMULA
| 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 (callan(AT)stat.wisc.edu), Aug 01 2004
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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);
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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, 1, 20}] - Robert A. Russell (russell(AT)post.harvard.edu), Dec 11 2004
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CROSSREFS
| Cf. A007282, A057162.
Sequence in context: A010364 A110391 A197460 * A053892 A013126 A012969
Adjacent sequences: A001680 A001681 A001682 * A001684 A001685 A001686
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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