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Search: id:A001683
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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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+0
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; listen)
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OFFSET
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2,5
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COMMENT
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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
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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.
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
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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
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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
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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
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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
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Cf. A007282, A057162.
Sequence in context: A064035 A010364 A110391 this_sequence A053892 A013126 A012969
Adjacent sequences: A001680 A001681 A001682 this_sequence A001684 A001685 A001686
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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