%I A000207 M2375 N0942
%S A000207 1,1,1,3,4,12,27,82,228,733,2282,7528,24834,83898,285357,983244,
%T A000207 3412420,11944614,42080170,149197152,531883768,1905930975,6861221666,
%U A000207 24806004996,90036148954,327989004892,1198854697588,4395801203290
%N A000207 Number of inequivalent ways of dissecting a regular (n+2)-gon into n 
               triangles by n-1 non-intersecting diagonals under rotations and reflections; 
               also the number of planar 2-trees.
%C A000207 Also a(n) is the number of hexaflexagons of order n+2. - Mike Godfrey 
               (m.godfrey(AT)umist.ac.uk), Feb 25 2002 (see the Kosters paper).
%D A000207 L. W. Beineke and R. E. Pippert, Enumerating labeled k-dimensional trees 
               and ball dissections, pp. 12-26 of Proceedings of Second Chapel Hill 
               Conference on Combinatorial Mathematics and Its Applications, University 
               of North Carolina, Chapel Hill, 1970. Reprinted in Math. Annalen, 
               191 (1971), 87-98.
%D A000207 B. N. Cyvin, E. Brendsdal, J. Brunvoll and S. J. Cyvin, Isomers of polyenes 
               attached to benzene, Croatica Chemica Acta, 68 (1995), 63-73.
%D A000207 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.
%D A000207 R. K. Guy, ``Dissecting a polygon into triangles,'' Bull. Malayan Math. 
               Soc., Vol. 5, pp. 57-60, 1958.
%D A000207 F. Harary and E. M. Palmer, On acyclic simplicial complexes. Mathematika 
               15 1968 115-122.
%D A000207 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 
               1973, p. 79, Table 3.5.1 (the entries for n=16 and n=21 appear to 
               be incorrect).
%D A000207 F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for 
               arbitrary polygons, Discr. Math. 11 (1975), 371-389 (the entries 
               for n=4 and n=30 appear to be incorrect).
%D A000207 M. Kosters, A theory of hexaflexagons, Nieuw Archief Wisk., 17 (1999), 
               349-362.
%D A000207 J. W. Moon and L. Moser, Triangular dissections of n-gons, Canad. Math. 
               Bull., 6 (1963), 175-178.
%D A000207 T. S. Motzkin, Relations between hypersurface cross ratios and a combinatorial 
               formula for partitions of a polygon, for permanent preponderance 
               and for non-associative products, Bull. Amer. Math. Soc., 54 (1948), 
               352-360 (the entry for n=10 appears to be incorrect).
%D A000207 C. O. Oakley and R. J. Wisner, Flexagons, Amer. Math. Monthly, 64 (1957), 
               143-154.
%D A000207 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000207 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000207 P. K. Stockmeyer, The charm bracelet problem and its applications, pp. 
               339-349 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by 
               R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 
               1974.
%H A000207 T. D. Noe, <a href="b000207.txt">Table of n, a(n) for n=1..200</a>
%H A000207 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Sequences realized by oligomorphic permutation groups</a>, J. Integ. 
               Seqs. Vol. 3 (2000), #00.1.5.
%H A000207 A. S. Conrad and D. K. Hartline, <a href="http://theory.lcs.mit.edu/~edemaine/
               flexagons/Conrad-Hartline-1962/flexagon.html">Flexagons</a>
%H A000207 Len Smiley, <a href="a000207.jpg">Illustration of initial terms</a>
%F A000207 a(n) = C(n)/(2*n)+C(n/2+1)/4+C(k)/2+C(n/3+1)/3 where C(n) = A000108(n-2) 
               if n is an integer, 0 otherwise and k = (n+1)/2 if n is odd, k = 
               n/2+1 if n is even. Thus C(2), C(3), C(4), C(5), ... are 1, 1, 2, 
               5, ...
%F A000207 G.f.=[12(1+x-2x^2)+(1-4x)^(3/2)-3(3+2x)(1-4x^2)^(1/2)-4(1-4x^3)^(1/2)]/
               (24x^2). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 19 2004, 
               from the S. J. Cyvin et al. reference.
%F A000207 a(n) ~ A000108(n)/(2*n+4) ~ 4^n / (2 sqrt(n Pi)*(n + 1)*(n + 2)) [From 
               M. F. Hasler (MHasler(AT)univ-ag.fr), Apr 19 2009]
%e A000207 E.g. a 4-gon (n=2) could have either diagonal drawn, C(3)=2, but with 
               essentially only one result. A 5-gon (n=3) gives C(4)=5, but they 
               each have 2 diags emanating from 1 of the 5 vertices and are essentially 
               the same. A 6-gon can have a nuclear disarmament sign (6 ways), an 
               N (3 ways and 3 reflexions) or a triangle (2 ways) of diagonals, 
               6 + 6 + 2 = 14 = C(5), but only 3 essentially different.a - R. K. 
               Guy, Mar 06 2004
%p A000207 A000108 := proc(n) if n >= 0 then binomial(2*n,n)/(n+1) ; else 0; fi; 
               end:
%p A000207 A000207 := proc(n) option remember: local k, it1, it2;
%p A000207 if n mod 2 = 0 then k := n/2+2 else k := (n+3)/2 fi:
%p A000207 if n mod 2 <> 0 then it1 := 0 else it1 := 1 fi:
%p A000207 if (n+2) mod 3 <> 0 then it2 := 0 else it2 := 1 fi:
%p A000207 RETURN(A000108(n)/(2*n+4) + it1*A000108(n/2)/4 + A000108(k-2)/2 + it2*A000108((n-1)/
               3)/3)
%p A000207 end:
%p A000207 seq(A000207(n),n=1..30) ; (Revised Maple program from R. J. Mathar, Apr 
               19 2009)
%p A000207 A000207 := proc(n) option remember: local k,it1,it2; if n mod 2 = 0 then 
               k := n/2+1 else k := (n+1)/2 fi: if n mod 2 <> 0 then it1 := 0 else 
               it1 := 1 fi: if n mod 3 <> 0 then it2 := 0 else it2 := 1 fi: RETURN(A000108(n-2)/
               (2*n) + it1*A000108(n/2+1-2)/4 + A000108(k-2)/2 + it2*A000108(n/3+1-2)/
               3) end:
%p A000207 A000207 := n->(A000108(n)/(n+2)+A000108(floor(n/2))*((1+(n+1 mod 2) /
               2)))/2+`if`(n mod 3=1,A000108(floor((n-1)/3))/3,0); # [From P. Luschny, 
               Apr 19 2009 and M. F. Hasler (MHasler(AT)univ-ag.fr), Apr 19 2009]
%p A000207 G:=(12*(1+x-2*x^2)+(1-4*x)^(3/2)-3*(3+2*x)*(1-4*x^2)^(1/2)-4*(1-4*x^3)^(1/
               2))/24/x^2: Gser:=series(G,x=0,35): seq(coeff(Gser,x^n),n=1..31); 
               (Deutsch)
%t A000207 p=3; Table[(Binomial[(p-1)n, n]/(((p-2)n+1)((p-2)n+2)) + If[OddQ[n], 
               If[OddQ[p], Binomial[(p-1)n/2, (n-1)/2]/n, (p+1)Binomial[((p-1)n-1)/
               2, (n-1)/2]/((p-2)n+2)], 3Binomial[(p-1)n/2, n/2]/((p-2)n+2)]+Plus 
               @@ Map[EulerPhi[ # ]Binomial[((p-1)n+1)/#, (n-1)/# ]/((p-1)n+1)&, 
               Complement[Divisors[GCD[p, n-1]], {1, 2}]])/2, {n, 1, 20}] - Robert 
               A. Russell (russell(AT)post.harvard.edu), Dec 11 2004
%o A000207 (PARI) A000207(n)=(A000108(n)/(n+2)+A000108(n\2)*if(n%2,1,3/2))/2+if(n%3==1,
               A000108(n\3)/3) - M. F. Hasler (MHasler(AT)univ-ag.fr), Apr 19 2009
%Y A000207 Cf. A000577, A070765.
%Y A000207 Sequence in context: A000577 A111758 A000942 this_sequence A002986 A147569 
               A090660
%Y A000207 Adjacent sequences: A000204 A000205 A000206 this_sequence A000208 A000209 
               A000210
%K A000207 nonn,nice,easy
%O A000207 1,4
%A A000207 N. J. A. Sloane (njas(AT)research.att.com).
%E A000207 More terms from James Sellers, Jul 10 2000