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%I #35 Jun 04 2021 09:46:07
%S 0,1,0,1,3,0,0,6,3,20,0,0,29,0,0,56,101,108,0,392
%N Number of grid-filling curves of order n (on the square grid) with turns by +-90 degrees generated by folding morphisms that are perfect.
%C 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 starting with D, 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 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. Grid-filling or plane-filling means that all the points in arbitrary large balls of gridpoints are eventually visited by the Cj. Perfect means that four 90-degree rotated copies of the curves Cj started at the origin will pass exactly twice through all grid-points as j tends to infinity (except the origin itself).
%C It is a theorem that a(A022544(n)) = 0, and a(A001481(n)) > 0 for n>2.
%D 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.
%H F. M. Dekking, <a href="https://doi.org/10.1016/j.tcs.2011.09.025">Paperfolding Morphisms, Planefilling Curves, and Fractal Tiles</a>, Theoretical Computer Science, volume 414, issue 1, January 2012, pages 20-37. Also <a href="http://arxiv.org/abs/1011.5788">arXiv:1011.5788</a> [math.CO], 2010-2011.
%e 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).
%Y Cf. A296148, A343990, A343991.
%K nonn,more
%O 1,5
%A _N. J. A. Sloane_, May 06 2021
%E Renamed and rewritten by _Michel Dekking_, Jun 03 2021
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