This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A343990 #43 Dec 30 2024 01:16:43
%S A343990 0,0,1,1,2,7,10,15,33,45,93,186,300,530,825,1561,2722,4685,7419,13563
%N A343990 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.
%C A343990 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.
%C A343990 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.
%D A343990 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).
%H A343990 Michel Dekking, <a href="/A343990/b343990.txt">Table of n, a(n) for n = 1..20</a>
%H A343990 Chandler Davis and Donald E. Knuth, <a href="/A005811/a005811.pdf">Number Representations and Dragon Curves</a>, Journal of Recreational Mathematics, volume 3, number 2, April 1970, pages 66-81, and number 3, July 1970, pages 133-149. [Cached copy, with permission]
%H A343990 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 A343990 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.
%Y A343990 Cf. A296148, A343991, A343992.
%K A343990 nonn,more
%O A343990 1,5
%A A343990 _N. J. A. Sloane_, May 06 2021
%E A343990 Rewritten and renamed by _Michel Dekking_, Jun 06 2021