A343991 Number of grid-filling curves of order n (on the square grid) with turns by +-90 degrees generated by folding morphisms that are plane-filling but not perfect.

0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 8, 0, 4, 0, 12

Offset: 1

Author

N. J. A. Sloane, May 06 2021

Keywords

• nonn
• more

Comments

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.

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).

It is a theorem that a(A022544(n)) = 0 for all n.

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 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.

Links

• Michel Dekking, Table of n, a(n) for n = 1..20
• Chandler Davis and Donald E. Knuth, Number Representations and Dragon Curves, 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]
• F. M. Dekking, Paperfolding Morphisms, Planefilling Curves, and Fractal Tiles, Theoretical Computer Science, volume 414, issue 1, January 2012, pages 20-37. Also arXiv:1011.5788 [math.CO], 2010-2011.

Crossrefs

Cf. A296148, A343990, A343992.

Extensions

Rewritten and renamed by Michel Dekking, Jun 06 2021