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A000063
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Symmetrical dissections of an n-gon.
(Formerly M0978 N0367)
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3
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1, 1, 2, 4, 5, 14, 14, 39, 42, 132, 132, 424, 429, 1428, 1430, 4848, 4862, 16796, 16796, 58739, 58786, 208012, 208012, 742768, 742900, 2674426, 2674440, 9694416, 9694845, 35357670, 35357670, 129643318, 129644790, 477638700, 477638700, 1767258328, 1767263190, 6564120288
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OFFSET
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5,3
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COMMENTS
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This sequence, S_n in Guy's 1958 paper, counts triangulations of a regular n-gon into n-2 triangles such that the only symmetries of the triangulation are the identity and a single reflection ("symmetry of a kite"). 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(floor(n/2) - 1) - Catalan(n/4 - 1) - Catalan (n/6 - 1), where Catalan(x) = 0 for noninteger x (from Guy's 1958 paper). - Joseph Myers, Jun 21 2012
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MATHEMATICA
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c[n_Integer] := CatalanNumber[n]; c[_] = 0; a[n_] := c[Floor[n/2]-1] - c[n/4-1] - c[n/6-1]; Array[a, 40, 5] (* Jean-François Alcover, Feb 03 2016, after Joseph Myers *)
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PROG
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(PARI)
C(n)=if(type(n)==type(1), binomial(2*n, n)/(n+1), 0);
a(n)=C(floor(n/2)-1) - C(n/4-1) - C(n/6-1);
vector(66, n, a(n+4))
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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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