A265671 Directions of edges in a plane-filling curve of order 13.

1, 2, 2, 2, 1, 3, 1, 1, 2, 3, 2, 2, 1, 2, 3, 3, 3, 2, 1, 2, 2, 3, 1, 3, 3, 2, 2, 3, 3, 3, 2, 1, 2, 2, 3, 1, 3, 3, 2, 2, 3, 3, 3, 2, 1, 2, 2, 3, 1, 3, 3, 2, 1, 2, 2, 2, 1, 3, 1, 1, 2, 3, 2, 2, 1, 3, 1, 1, 1, 3, 2, 3, 3, 1, 2, 1, 1, 3, 1, 2, 2, 2, 1, 3, 1, 1, 2, 3, 2, 2, 1, 1, 2, 2, 2, 1, 3, 1, 1, 2, 3, 2, 2, 1, 2, 3, 3

Offset

1,2

Comments

Infinite ternary word generated from the axiom 1 by the Lindenmayer system with maps 1 --> 1222131123221, 2 --> 2333212231332, and 3 --> 3111323312113.

This is a 13-automatic sequence. It can be generated by reading the lowest nonzero digit D in the base-13 expansion of n>=1: a(n)=1 for D \in {1, 5, 7, 8}, a(n)=2 for D \in {2, 3, 4, 9, 11, 12}, and a(n)=3 for D \in {6, 10}.

Corresponds to a grid-filling curve on the triangular grid as a sequence of directed edges where the letters are the directions of the third roots of unity. See the file titled "First iterate of the curve".

The corresponding sequence of turns (by 0 or +-120 degree) can be obtained from the L-system with axiom + and maps + --> +00--+0++-0-+, 0 --> +00--+0++-0-0, and - --> +00--+0++-0--.

The shape of the curve is one of the A234434(13)=15 possible shapes.

An L-system with axiom F and just one non-constant map F --> F+F0F0F-F-F+F0F+F+F-F0F-F generates the curve when 0, +, and - are interpreted as turns and F as a unit stroke in the current direction.

Three copies of the curve can be arranged to create a rep-tile that is a lattice tiling, see the files "Tile-plus" (axiom F+F+F), "Tile-minus" (Axiom F-F-F), "Tiling-plus" (self-similarity of the Tile-plus), and "Complex numeration system" (giving the generalized unit square of a numeration system with base 1 + i * sqrt(12) that reproduces the Tile-plus).

Links

Joerg Arndt, Table of n, a(n) for n = 1..2197

Joerg Arndt, First iterate of the curve, Second iterate, Third iterate, Fourth iterate.

Joerg Arndt, Rendering used for the T-shirt on Neil's 75th birthday (png image, 1716 X 2732 pixel).

Joerg Arndt, Tile-plus, Tile-minus, Tiling-plus, Complex numeration system.

Joerg Arndt, Plane-filling curves on all uniform grids, arXiv:1607.02433 [math.CO], (8-July-2016).

Jörg Arndt and Julia Handl, Edge-covering plane-filling curves on grid colorings: a pedestrian approach, arXiv:2312.00654 [math.CO].

Index entries for 13-automatic sequences.

Mathematica

SubstitutionSystem[{1 -> {1, 2, 2, 2, 1, 3, 1, 1, 2, 3, 2, 2, 1}, 2 -> {2, 3, 3, 3, 2, 1, 2, 2, 3, 1, 3, 3, 2}, 3 -> {3, 1, 1, 1, 3, 2, 3, 3, 1, 2, 1, 1, 3}}, {1}, {2}][[1]] (* Paolo Xausa, Jun 11 2024 *)

Crossrefs

Cf. A029883, A035263, A060236, A080846, A156595, A175337, A176405, A176416.

Cf. A234434 (curves on the triangular grid).

Cf. A229214 (a similar L-system for Gosper's flowsnake).

Sequence in context: A308367 A205599 A327891 * A214707 A083039 A106253

Adjacent sequences: A265668 A265669 A265670 * A265672 A265673 A265674

Keyword

nonn

Author

Joerg Arndt, Dec 13 2015

Status

approved