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 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. 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 (list; graph; refs; listen; history; text; internal format)
 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

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