A234434 -id:A234434 - cp's OEIS frontend

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.

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

Joerg Arndt, Dec 13 2015

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

Joerg Arndt, Plane-filling curves on all uniform grids, arXiv preprint arXiv:1607.02433 [math.CO], 2016.
Jörg Arndt, Julia Handl, Plane-filling Folding Curves on the Square Grid, 2018. (This paper is about a different class of curves where the Lindenmayer-systems have two no-constant symbols)

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).

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

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

Original entry on oeis.org

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

Joerg Arndt, Dec 13 2015

nonn

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).

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.

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).

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 *)

A265686 Number of shapes of grid-filling curves of order 6*n+1 (on the tri-hexagonal grid) with turns by +-60 and +-120 degrees that are generated by Lindenmayer-systems with just one symbol apart from the turns.

Original entry on oeis.org

1, 3, 7, 10, 63, 157, 456, 1830, 0, 8538, 23114, 61804, 165123, 0, 2339000

Offset: 1

Joerg Arndt, Dec 13 2015

Such curves exist only for n such that 6*n+1 is a term of A003136, these values 6*n+1 are given in A260682.

The first such curve, the only one of order 7, was discovered by Jeffrey Ventrella, see page 105 in the Ventrella reference.

Joerg Arndt, Plane-filling curves on all uniform grids, arXiv preprint arXiv:1607.02433 [math.CO], 2016.
Jeffrey J. Ventrella, Brain-Filling Curves: A Fractal Bestiary, LuLu.com, (2012).

Cf. A234434 (shapes on the triangular grid), A265685 (shapes on the square grid).

a(11) - a(15) from Joerg Arndt, Feb 10 2019

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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.

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

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A265686 Number of shapes of grid-filling curves of order 6*n+1 (on the tri-hexagonal grid) with turns by +-60 and +-120 degrees that are generated by Lindenmayer-systems with just one symbol apart from the turns.

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