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 A000063 Symmetrical dissections of an n-gon. (Formerly M0978 N0367) 3
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 5,3 COMMENTS 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 A000108 is a subsequence, see formula. - Ralf Stephan, Aug 19 2004 (edited, Joerg Arndt, Aug 31 2014) REFERENCES 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). LINKS Joseph Myers, Table of n, a(n) for n = 5..1000 S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy] R. K. Guy, Dissecting a polygon into triangles, Research Paper #9, Math. Dept., Univ. Calgary, 1967. [Annotated scanned copy] FORMULA a(2n+3) = A000108(n), n>0. - M. F. Hasler, Mar 25 2012 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 MATHEMATICA 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 *) PROG (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)) \\ Joerg Arndt, Aug 31 2014 CROSSREFS Sequence in context: A127077 A104549 A174513 * A039574 A182375 A295402 Adjacent sequences:  A000060 A000061 A000062 * A000064 A000065 A000066 KEYWORD nonn AUTHOR EXTENSIONS Extended by Joseph Myers, Jun 21 2012 STATUS approved

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Last modified August 18 12:03 EDT 2019. Contains 326090 sequences. (Running on oeis4.)