

A234434


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


4



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
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OFFSET

3,5


COMMENTS

Shapes are considered modulo reflections and rotations.
The curves considered are not selfintersecting, not edgecontacting (i.e., have double edges), but (necessarily) vertexcontacting (i.e., a point in the grid is visited more than once).
The Lsystems are interpreted as follows: 'F' is a unitstroke 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 3symmetric, and sometimes (only for n of the form 6*k+1) 6symmetric. There could in general be more tileshapes than curveshapes, for n=7 both cardinalities coincide, see links section. It turns out that for large n there are actually fewer tileshapes than curveshapes.
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 Lsystems, 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


LINKS

Table of n, a(n) for n=3..62.
Joerg Arndt, Matters Computational (The Fxtbook), see section 1.31.5 "Dragon curves based on radixR counting", pp. 95101, images of the R7dragons are given on p. 97 and p. 98
Joerg Arndt, all 3 shapes of curves of order 7, rendered after 4 generations of the Lsystems.
Joerg Arndt, all 3 shapes of tiles of order 7, rendered after 4 generations of the Lsystems, curves colored to make them apparent.
Joerg Arndt, all 15 shapes of curves of order 13, rendered after 3 generations of the Lsystems (file size about 500 kB).
Joerg Arndt, all shapes of tiles of order 13, rendered after 3 generations of the Lsystems (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 order13 curves into selfsimilar parts (file size about 1.3 MB)
Joerg Arndt, Planefilling curves on all uniform grids, arXiv preprint arXiv:1607.02433 [math.CO], 2016.
Jeffrey J. Ventrella, BrainFilling Curves: A Fractal Bestiary, 2012.


EXAMPLE

The a(3)=1 shape of order 3 is generated by F > F+FF, the curve generated by F > FF+F has the same shape (after reflection). The curve is called the "terdragon", see A080846.
There are 5 Lsystems that generate a curve of order 7 with first turn '0' or '+':
F > F0F+F0FFF+F # R71
F > F0F+F+FFF0F # R72
F > F+F0F+FF0FF # R73
F > F+FFF0F+F0F # R74 # same shape as R71
F > F+FFF+F+FF # R75 # same shape as R72
As shown, these give just 3 shapes (and the Lsystems with first turn '' give no new shapes), so a(7)=3.
The curve R71 appears on page 107 in the Ventrella reference.
The symmetric curves R72 and R75 appear in the Arndt reference (there named "R7dragon" and "second R7dragon", see A176405 and A176416).


CROSSREFS

Cf. A265685 (shapes on the square grid), A265686 (trihexagonal grid).
Sequence in context: A227498 A131986 A002656 * A234020 A276833 A166586
Adjacent sequences: A234431 A234432 A234433 * A234435 A234436 A234437


KEYWORD

nonn,hard,more,nice


AUTHOR

Joerg Arndt, Dec 26 2013


EXTENSIONS

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


STATUS

approved



