

A000063


Symmetrical dissections of an ngon.
(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
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OFFSET

5,3


COMMENTS

This sequence, S_n in Guy's 1958 paper, counts triangulations of a regular ngon into n2 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 ngon do not count as distinct.  Joseph Myers, Jun 21 2012


REFERENCES

R. K. Guy, Dissecting a polygon into triangles, Bull. Malayan Math. Soc., Vol. 5, pp. 5760, 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



FORMULA

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/41]  c[n/61]; Array[a, 40, 5] (* JeanFrançois 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/41)  C(n/61);
vector(66, n, a(n+4))


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



