

A343992


Number of gridfilling curves of order n (on the square grid) with turns by +90 degrees generated by folding morphisms that are perfect.


2



0, 1, 0, 1, 3, 0, 0, 6, 3, 20, 0, 0, 29, 0, 0, 56, 101, 108, 0, 392
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OFFSET

1,5


COMMENTS

Curves of order n generated by folding morphisms are walks on the square grid, also coded by sequences (starting with D) of n1 U's and D's starting with D, the Up and Down folds. These are also known as nfolds. 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 sigma(a) = 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. Gridfilling or planefilling means that all the points in arbitrary large balls of gridpoints are eventually visited by the Cj. Perfect means that four 90degree rotated copies of the curves Cj started at the origin will pass exactly twice through all gridpoints as j tends to infinity (except the origin itself).
It is a theorem that a(A022544(n)) = 0, and a(A001481(n)) > 0 for n>2.


REFERENCES

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 6681, and number 3, July 1970, pages 133149. Reprinted and updated in Donald E. Knuth, Selected Papers on Fun and Games, CSLI Publications, 2010, pages 571614.


LINKS

Table of n, a(n) for n=1..20.
F. M. Dekking, Paperfolding Morphisms, Planefilling Curves, and Fractal Tiles, Theoretical Computer Science, volume 414, issue 1, January 2012, pages 2037. Also arXiv:1011.5788 [math.CO], 20102011.


EXAMPLE

For n=2 one obtains Heighway's dragon curve, with folding morphism sigma: a>ab, b>cb, c> cd, d> ad (see A105500 or A246960).


CROSSREFS

Cf. A296148, A343990, A343991.
Sequence in context: A135028 A325006 A325014 * A275689 A322015 A285311
Adjacent sequences: A343989 A343990 A343991 * A343993 A343994 A343995


KEYWORD

nonn,more


AUTHOR

N. J. A. Sloane, May 06 2021


EXTENSIONS

Renamed and rewritten by Michel Dekking, Jun 03 2021


STATUS

approved



