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A343990
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Number of grid-filling curves of order n (on the square grid) with turns by +-90 degrees generated by folding morphisms that are self-avoiding but not plane-filling.
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3
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0, 0, 1, 1, 2, 7, 10, 15, 33, 45, 93, 186, 300, 530, 825, 1561, 2722, 4685, 7419, 13563
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OFFSET
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1,5
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COMMENTS
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Curves of order n generated by folding morphisms are walks on the square grid, also coded by sequences (starting with D) of n-1 U's and D's, the Up and Down folds. These are also known as n-folds. In the square grid they uniquely correspond to folding morphisms, which are a special class of morphisms sigma on the alphabet {a,b,c,d}. (There is in particular the requirement that the first two letters of sigma(a) are ab.) Here the letters a,b,c, and d correspond to the four possible steps of the walk.
A curve C = C1 of order n generates curves Cj of order n^j by the process of iterated folding. Iterated folding corresponds to iterates of the folding morphism. Grid-filling or plane-filling means that all the points in arbitrary large balls of gridpoints are eventually visited by the Cj.
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REFERENCES
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Chandler Davis and Donald E. Knuth, Number Representations and Dragon Curves -- I and II, Journal of Recreational Mathematics, volume 3, number 2, April 1970, pages 66-81, and number 3, July 1970, pages 133-149. Reprinted and updated in Donald E. Knuth, Selected Papers on Fun and Games, CSLI Publications, 2010, pages 571-614. See page 611, table A_s = a(s).
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LINKS
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EXAMPLE
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Examples for n = 5 are given in Knuth's 2010 update. There are pictures which show (or suggest) that the 5-folds coded by DDUU, DUDD, DDUD are perfect, DUUD and DUDU yield a self-avoiding curve which is not plane-filling, and the other 3 give self-intersecting curves. So A343992(5) = 3 and a(5) = 2.
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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