This site is supported by donations to The OEIS Foundation.

 Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS". Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A000131 Number of asymmetrical dissections of n-gon. (Formerly M1535 N0599) 3
 2, 5, 21, 61, 214, 669, 2240, 7330, 24695, 83257, 284928, 981079, 3410990, 11937328, 42075242, 149171958, 531866972, 1905842605, 6861162880, 24805692978, 90035940227, 327987890608, 1198853954688, 4395797189206, 16165195705544, 59609156824273, 220373268471398, 816677398144221 (list; graph; refs; listen; history; text; internal format)
 OFFSET 7,1 COMMENTS This sequence, U_n in Guy's 1958 paper, counts triangulations of a regular n-gon into n-2 triangles with no nonidentity symmetries.  Triangulations related by a symmetry of the underlying n-gon 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. 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 = 7..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(n) = (Catalan(n-2) - (n/2)*Catalan(n/2 - 1) - n*Catalan(floor(n/2) - 1) - (n/3)*Catalan(n/3 - 1) + n*Catalan(n/4 - 1) + n*Catalan(n/6 - 1))/(2*n), where Catalan(x) = 0 for noninteger x (derived from Guy's 1958 paper). - Joseph Myers, Jun 21 2012 MATHEMATICA catalan[n_] := Block[{c = Binomial[2 n, n]/(n + 1)}, If[IntegerQ[c], c, 0]]; f[n_] := (catalan[n - 2] - (n/2) catalan[n/2 - 1] - n*catalan[Floor[n/2] - 1] - (n/3)*catalan[n/3 - 1] + n*catalan[n/4 - 1] + n*catalan[n/6 - 1])/(2 n); Array[f, 28, 7] (* Robert G. Wilson v, Jun 23 2014 *) PROG (PARI) C(n)=if(denominator(n)==1, binomial(2*n, n)/(n+1), 0) a(n)=(C(n-2)/n-C(n/2-1)/2-C(n\2-1)-C(n/3-1)/3+C(n/4-1)+C(n/6-1))/2 \\ Charles R Greathouse IV, Apr 05 2013 CROSSREFS Cf. A000063. Sequence in context: A039777 A246167 A000941 * A242785 A228385 A152801 Adjacent sequences:  A000128 A000129 A000130 * A000132 A000133 A000134 KEYWORD nonn AUTHOR EXTENSIONS Extended by Joseph Myers, Jun 21 2012 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.