cp's OEIS Frontend

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.

Showing 1-6 of 6 results.

A265685 Number of shapes of grid-filling curves of order 4*n+1 (on the square grid) with turns by +-90 degrees that are generated by Lindenmayer-systems with just one symbol apart from the turns.

Original entry on oeis.org

1, 1, 4, 6, 0, 33, 39, 0, 164, 335, 0, 603, 2467, 0, 10412, 19692, 0, 79494, 0, 155155, 1271455, 1272243, 0
Offset: 1

Views

Author

Joerg Arndt, Dec 13 2015

Keywords

Comments

Such curves exist only for n such that 4*n+1 is a term of A057653.

Links

Crossrefs

Cf. A234434 (shapes on the triangular grid), A265686 (tri-hexagonal grid).
Cf. A296148 (folding curves of order n) and A296149 (folding curves of order 4*n+1).
Cf. A306358 (curve orders with at least two decomposition x^2 + y^2).

Extensions

a(15)..a(23) from Joerg Arndt, Feb 12 2019

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.

Original entry on oeis.org

0, 0, 1, 1, 2, 7, 10, 15, 33, 45, 93, 186, 300, 530, 825, 1561, 2722, 4685, 7419, 13563
Offset: 1

Views

Author

N. J. A. Sloane, May 06 2021

Keywords

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.

Examples

			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.

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. See page 611, table A_s = a(s).

Links

Crossrefs

Cf. A296148, A343991, A343992.

Extensions

Rewritten and renamed by Michel Dekking, Jun 06 2021

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.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 8, 0, 4, 0, 12
Offset: 1

Views

Author

N. J. A. Sloane, May 06 2021

Keywords

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

Crossrefs

Cf. A296148, A343990, A343992.

Extensions

Rewritten and renamed by Michel Dekking, Jun 06 2021

A343992 Number of grid-filling curves of order n (on the square grid) with turns by +-90 degrees generated by folding morphisms that are perfect.

Original entry on oeis.org

0, 1, 0, 1, 3, 0, 0, 6, 3, 20, 0, 0, 29, 0, 0, 56, 101, 108, 0, 392
Offset: 1

Views

Author

N. J. A. Sloane, May 06 2021

Keywords

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 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).
It is a theorem that a(A022544(n)) = 0, and a(A001481(n)) > 0 for n>2.

Examples

			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).

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

Crossrefs

Cf. A296148, A343990, A343991.

Extensions

Renamed and rewritten by Michel Dekking, Jun 03 2021

A296147 Number of shapes of grid-filling curves of order A001481(n) (on the square grid) with turns by +-90 degrees that are generated by folding morphisms.

Original entry on oeis.org

1, 1, 1, 2, 2, 6, 3, 20, 14, 44, 32, 69, 212, 287, 796, 438, 1402, 4232, 3202, 2242, 14316, 5080, 11122, 12374, 155305, 152602, 77469
Offset: 1

Views

Author

Joerg Arndt and Julia Handl, Dec 06 2017

Keywords

Comments

a(1) and a(2) correspond to the trivial (empty and single-stroke) curves of orders 0 and 1 respectively.

Links

Crossrefs

Cf. A296148 (same sequence, including zero terms).
Cf. A265685 (simple curves of order 4*n+1).

A296149 Number of shapes of grid-filling curves of order 4*n+1 (on the square grid) with turns by +-90 degrees that are generated by folding morphisms.

Original entry on oeis.org

2, 3, 14, 32, 0, 287, 438, 0, 2242, 5080, 11122, 12374, 77469
Offset: 1

Views

Author

Joerg Arndt, Dec 06 2017

Keywords

Comments

Terms are nonzero if and only if 4*n+1 is a term of A057653.

Links

Crossrefs

Cf. A296148 (number of folding curves of all orders).
Cf. A265685 (simple curves of order 4*n+1).
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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