

A156209


Number of possible values of C(v) = the number of valid mountainvalley assignments for a flatfoldable origami vertex v of degree 2n.


0



1, 3, 7, 13, 24, 39, 62, 97, 147, 215, 312, 440, 617, 851, 1161, 1569, 2098, 2778, 3649, 4764, 6163, 7939, 10160, 12924, 16361, 20613, 25833, 32259, 40097, 49667, 61272, 75337, 92306, 112755, 137272, 166654, 201734, 243582, 293288
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OFFSET

1,2


LINKS

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), 2938.


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



