%N 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.
%C Shapes are considered modulo reflections and rotations.
%C 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).
%C 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".
%C The images in the links section use rounded corners to make the curves visually better apparent.
%C 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.
%C Terms a(n) are nonzero for n>=3 if and only if n is a term of A003136.
%C The equivalent sequence for the square grid has nonzero terms for n>=5 that are terms of A057653.
%C 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.
%C 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
%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, 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
%H Joerg Arndt, <a href="/A234434/a234434.pdf">all 3 shapes of curves of order 7</a>, rendered after 4 generations of the L-systems.
%H Joerg Arndt, <a href="/A234434/a234434_1.pdf">all 3 shapes of tiles of order 7</a>, rendered after 4 generations of the L-systems, curves colored to make them apparent.
%H Joerg Arndt, <a href="/A234434/a234434_2.pdf">all 15 shapes of curves of order 13</a>, rendered after 3 generations of the L-systems (file size about 500 kB).
%H Joerg Arndt, <a href="/A234434/a234434_3.pdf">all shapes of tiles of order 13</a>, 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).
%H Joerg Arndt, <a href="/A234434/a234434_4.pdf">decompositions of order-13 curves into self-similar parts</a> (file size about 1.3 MB)
%H Joerg Arndt, <a href="https://arxiv.org/abs/1607.02433">Plane-filling curves on all uniform grids</a>, arXiv preprint arXiv:1607.02433 [math.CO], 2016.
%H Jeffrey J. Ventrella, <a href="http://archive.org/details/BrainfillingCurves-AFractalBestiary">Brain-Filling Curves: A Fractal Bestiary</a>, 2012.
%e 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.
%e There are 5 L-systems that generate a curve of order 7 with first turn '0' or '+':
%e F |--> F0F+F0F-F-F+F # R7-1
%e F |--> F0F+F+F-F-F0F # R7-2
%e F |--> F+F0F+F-F0F-F # R7-3
%e F |--> F+F-F-F0F+F0F # R7-4 # same shape as R7-1
%e F |--> F+F-F-F+F+F-F # R7-5 # same shape as R7-2
%e As shown, these give just 3 shapes (and the L-systems with first turn '-' give no new shapes), so a(7)=3.
%e The curve R7-1 appears on page 107 in the Ventrella reference.
%e 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).
%Y Cf. A265685 (shapes on the square grid), A265686 (tri-hexagonal grid).
%A _Joerg Arndt_, Dec 26 2013
%E Terms a(21), a(27), a(28), and a(31) corrected by _Joerg Arndt_, Jun 20 2018
%E Terms a(32) - a(47) from _Joerg Arndt_, Jun 22 2018
%E Terms a(48) - a(51) from _Joerg Arndt_, Nov 18 2018
%E Terms a(52) - a(56) added and a(48) - a(49) corrected, _Joerg Arndt_, Apr 07 2019
%E Terms a(57) - a(62) from _Joerg Arndt_, Apr 10 2019