A176416 -id:A176416 - cp's OEIS frontend

A234434 Number of shapes of grid-filling curves (on the triangular grid) with turns by 0, +120, or -120 degrees that are generated by Lindenmayer-systems with just one symbol apart from the turns.

1, 1, 0, 0, 3, 0, 5, 0, 0, 10, 15, 0, 0, 17, 0, 0, 71, 0, 213, 0, 0, 0, 184, 0, 549, 845, 0, 0, 1850, 0, 0, 0, 0, 6700, 9787, 0, 30475, 0, 0, 0, 52184, 0, 0, 0, 0, 182043, 401377, 0, 0, 604809, 0, 0, 0, 0, 4318067, 0, 0, 0, 7158120, 0

Offset: 3

Joerg Arndt, Dec 26 2013

Shapes are considered modulo reflections and rotations.
The curves considered are not self-intersecting, not edge-contacting (i.e., have double edges), but (necessarily) vertex-contacting (i.e., a point in the grid is visited more than once).
The L-systems are interpreted as follows: 'F' is a unit-stroke in the current direction, '+' is a turn left by 120 degrees, '-' a turn right by 120 degrees, and '0' means "no turn".
The images in the links section use rounded corners to make the curves visually better apparent.
Three copies of each curve (connected by three turns '+' or three turns '-') give two tiles (that tile the triangular grid), but symmetric curves (any symmetry) give just one tile(-shape). The tiles are 3-symmetric, and sometimes (only for n of the form 6*k+1) 6-symmetric. There could in general be more tile-shapes than curve-shapes, for n=7 both cardinalities coincide, see links section. It turns out that for large n there are actually fewer tile-shapes than curve-shapes.
Terms a(n) are nonzero for n>=3 if and only if n is a term of A003136.
The equivalent sequence for the square grid has nonzero terms for n>=5 that are terms of A057653.
If more symbols are allowed for the L-systems, more curves are found, also if strokes of lengths other than one unit are allowed, see the Ventrella reference.
For n = 49 there are two pairs (x, y) such that x^2 + x*y + y^2 = n, (7, 0) and (5, 3), respectively giving 132271 and 269106 shapes (a(49) = 401377 = 132271 + 269106). The next n with two such pairs (x, y) is n = 91, with pairs (6, 5) and (9, 1) - Joerg Arndt, Apr 07 2019

			The a(3)=1 shape of order 3 is generated by F |--> F+F-F, the curve generated by F |--> F-F+F has the same shape (after reflection). The curve is called the "terdragon", see A080846.
There are 5 L-systems that generate a curve of order 7 with first turn '0' or '+':
F |--> F0F+F0F-F-F+F  #  R7-1
F |--> F0F+F+F-F-F0F  #  R7-2
F |--> F+F0F+F-F0F-F  #  R7-3
F |--> F+F-F-F0F+F0F  #  R7-4 # same shape as R7-1
F |--> F+F-F-F+F+F-F  #  R7-5 # same shape as R7-2
As shown, these give just 3 shapes (and the L-systems with first turn '-' give no new shapes), so a(7)=3.
The curve R7-1 appears on page 107 in the Ventrella reference.
The symmetric curves R7-2 and R7-5 appear in the Arndt reference (there named "R7-dragon" and "second R7-dragon", see A176405 and A176416).

Joerg Arndt, Matters Computational (The Fxtbook), see section 1.31.5 "Dragon curves based on radix-R counting", pp. 95-101, images of the R7-dragons are given on p. 97 and p. 98
Joerg Arndt, all 3 shapes of curves of order 7, rendered after 4 generations of the L-systems.
Joerg Arndt, all 3 shapes of tiles of order 7, rendered after 4 generations of the L-systems, curves colored to make them apparent.
Joerg Arndt, all 15 shapes of curves of order 13, rendered after 3 generations of the L-systems (file size about 500 kB).
Joerg Arndt, all shapes of tiles of order 13, rendered after 3 generations of the L-systems (file size about 500 kB). Note: not all symmetries are accounted for, so some tiles appear more than once (e.g., in flipped over form).
Joerg Arndt, decompositions of order-13 curves into self-similar parts (file size about 1.3 MB)
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, 2012.

Cf. A265685 (shapes on the square grid), A265686 (tri-hexagonal grid).

Terms a(21), a(27), a(28), and a(31) corrected by Joerg Arndt, Jun 20 2018
Terms a(32) - a(47) from Joerg Arndt, Jun 22 2018
Terms a(48) - a(51)  from Joerg Arndt, Nov 18 2018
Terms a(52) - a(56) added and a(48) - a(49) corrected, Joerg Arndt, Apr 07 2019
Terms a(57) - a(62) from Joerg Arndt, Apr 10 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 *)

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A234434 Number of shapes of grid-filling curves (on the triangular grid) with turns by 0, +120, or -120 degrees that are generated by Lindenmayer-systems with just one symbol apart from the turns.

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