A000063 Symmetrical dissections of an n-gon. (Formerly M0978 N0367)

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

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: A367027 A104549 A174513 * A368378 A367298 A039574

Adjacent sequences: A000060 A000061 A000062 * A000064 A000065 A000066

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Extended by Joseph Myers, Jun 21 2012

Status

approved