A265686 -id:A265686 - 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

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

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

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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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