1,4

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).

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.

B. N. Cyvin, E. Brendsdal, J. Brunvoll and S. J. Cyvin, Isomers of polyenes attached to benzene, Croatica Chemica Acta, 68 (1995), 63-73.

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.

R. K. Guy, ``Dissecting a polygon into triangles,'' Bull. Malayan Math. Soc., Vol. 5, pp. 57-60, 1958.

F. Harary and E. M. Palmer, On acyclic simplicial complexes. Mathematika 15 1968 115-122.

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).

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).

M. Kosters, A theory of hexaflexagons, Nieuw Archief Wisk., 17 (1999), 349-362.

J. W. Moon and L. Moser, Triangular dissections of n-gons, Canad. Math. Bull., 6 (1963), 175-178.

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).

C. O. Oakley and R. J. Wisner, Flexagons, Amer. Math. Monthly, 64 (1957), 143-154.

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.

T. D. Noe, Table of n, a(n) for n=1..200

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

A. S. Conrad and D. K. Hartline, Flexagons

Len Smiley, Illustration of initial terms

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, ...

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.

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

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:

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)

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

Adjacent sequences: A000204 A000205 A000206 this_sequence A000208 A000209 A000210

Sequence in context: A000577 A111758 A000942 this_sequence A002986 A090660 A000208

nonn,nice,easy

njas

More terms from James Sellers, Jul 10 2000