A156209 Number of possible values of C(v) = the number of valid mountain-valley assignments for a flat-foldable origami vertex v of degree 2n.

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

Offset

1,2

Links

Table of n, a(n) for n=1..39.

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

Sequence in context: A102948 A232533 A061263 * A076276 A296558 A309051

Adjacent sequences: A156206 A156207 A156208 * A156210 A156211 A156212

Keyword

nonn

Author

Thomas C. Hull (thull(AT)wnec.edu), Feb 05 2009

Status

approved