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%I M0978 N0367 #49 Feb 03 2016 16:02:42
%S 1,1,2,4,5,14,14,39,42,132,132,424,429,1428,1430,4848,4862,16796,
%T 16796,58739,58786,208012,208012,742768,742900,2674426,2674440,
%U 9694416,9694845,35357670,35357670,129643318,129644790,477638700,477638700,1767258328,1767263190,6564120288
%N Symmetrical dissections of an n-gon.
%C 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
%C A000108 is a subsequence, see formula. - _Ralf Stephan_, Aug 19 2004 (edited, _Joerg Arndt_, Aug 31 2014)
%D R. K. Guy, Dissecting a polygon into triangles, Bull. Malayan Math. Soc., Vol. 5, pp. 57-60, 1958.
%D R. K. Guy, Dissecting a polygon into triangles, Research Paper #9, Math. Dept., Univ. Calgary, 1967.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Joseph Myers, <a href="/A000063/b000063.txt">Table of n, a(n) for n = 5..1000</a>
%H S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, <a href="http://dx.doi.org/10.1021/ci00026a012">Enumeration of polyene hydrocarbons: a complete mathematical solution</a>, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751.
%H S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, <a href="/A002057/a002057.pdf">Enumeration of polyene hydrocarbons: a complete mathematical solution</a>, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy]
%H R. K. Guy, <a href="/A000108/a000108_11.pdf">Dissecting a polygon into triangles</a>, Research Paper #9, Math. Dept., Univ. Calgary, 1967. [Annotated scanned copy]
%F a(2n+3) = A000108(n), n>0. - _M. F. Hasler_, Mar 25 2012
%F 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
%t 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_ *)
%o (PARI)
%o C(n)=if(type(n)==type(1), binomial(2*n,n)/(n+1), 0);
%o a(n)=C(floor(n/2)-1) - C(n/4-1) - C(n/6-1);
%o vector(66,n, a(n+4))
%o \\ _Joerg Arndt_, Aug 31 2014
%K nonn
%O 5,3
%A _N. J. A. Sloane_
%E Extended by _Joseph Myers_, Jun 21 2012
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