OFFSET
1,2
LINKS
T. C. Hull, Counting Mountain-valley Assignments for Flat Folds, Ars Combinatorica, Vol. 67 (2003), 175-188.
T. C. Hull, The Combinatorics of Flat Folds: a Survey, in Origami^3: Proceedings of the Third International Meeting of Origami Science, Mathematics, and Education, A.K. Peters, Ltd., Natick, MA (2002), 29-38.
EXAMPLE
The n=1 case is degenerate; C(v) = 2 in this case, so a(1)=1. When n=2 we have a degree 4 vertex, and C(v) can take on the values 4, 6, or 8 depending on the angles between the creases, so a(2)=3. In the n=3 (degree 6) case, C(v) can be any of {8, 12, 16, 18, 20, 24, 30}. Thus a(3)=7. The possible values of C(v) can be determined by recursive equations found in (Hull, 2002, 2003).
MATHEMATICA
CK[1] = {2}; Print[Length[CK[1]]]; For[k = 2, k < 40, k++, CK[k] = Union[Flatten[Table[Union[Binomial[ 2 i, i]*CK[k - i], Binomial[2 i + 1, i]*CK[k - i]], {i, 1, k - 1}]], {2*Binomial[2*k, k - 1]}]; Print[Length[CK[k]]]]
CROSSREFS
KEYWORD
nonn
AUTHOR
Thomas C. Hull (thull(AT)wnec.edu), Feb 05 2009
STATUS
approved