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A000131
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Number of asymmetrical dissections of n-gon.
(Formerly M1535 N0599)
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3
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2, 5, 21, 61, 214, 669, 2240, 7330, 24695, 83257, 284928, 981079, 3410990, 11937328, 42075242, 149171958, 531866972, 1905842605, 6861162880, 24805692978, 90035940227, 327987890608, 1198853954688, 4395797189206, 16165195705544, 59609156824273, 220373268471398, 816677398144221
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OFFSET
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7,1
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COMMENTS
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This sequence, U_n in Guy's 1958 paper, counts triangulations of a regular n-gon into n-2 triangles with no nonidentity symmetries. Triangulations related by a symmetry of the underlying n-gon do not count as distinct. - Joseph Myers, Jun 21 2012
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REFERENCES
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R. K. Guy, Dissecting a polygon into triangles, Bull. Malayan Math. Soc., Vol. 5, pp. 57-60, 1958.
R. K. Guy, Dissecting a polygon into triangles, Research Paper #9, Math. Dept., Univ. Calgary, 1967.
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).
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LINKS
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FORMULA
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a(n) = (Catalan(n-2) - (n/2)*Catalan(n/2 - 1) - n*Catalan(floor(n/2) - 1) - (n/3)*Catalan(n/3 - 1) + n*Catalan(n/4 - 1) + n*Catalan(n/6 - 1))/(2*n), where Catalan(x) = 0 for noninteger x (derived from Guy's 1958 paper). - Joseph Myers, Jun 21 2012
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MATHEMATICA
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catalan[n_] := Block[{c = Binomial[2 n, n]/(n + 1)}, If[IntegerQ[c], c, 0]]; f[n_] := (catalan[n - 2] - (n/2) catalan[n/2 - 1] - n*catalan[Floor[n/2] - 1] - (n/3)*catalan[n/3 - 1] + n*catalan[n/4 - 1] + n*catalan[n/6 - 1])/(2 n); Array[f, 28, 7] (* Robert G. Wilson v, Jun 23 2014 *)
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PROG
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(PARI) C(n)=if(denominator(n)==1, binomial(2*n, n)/(n+1), 0)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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