%I
%S 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,
%T 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,
%U 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
%N Directions of edges in a planefilling curve of order 13.
%C Infinite ternary word generated from the axiom 1 by the Lindenmayer system with maps 1 > 1222131123221, 2 > 2333212231332, and 3 > 3111323312113.
%C This is a 13automatic sequence. It can be generated by reading the lowest nonzero digit D in the base13 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}.
%C Corresponds to a gridfilling 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".
%C The corresponding sequence of turns (by 0 or +120 degree) can be obtained from the Lsystem with axiom + and maps + > +00+0++0+, 0 > +00+0++00, and  > +00+0++0.
%C The shape of the curve is one of the A234434(13)=15 possible shapes.
%C An Lsystem with axiom F and just one nonconstant map F > F+F0F0FFF+F0F+F+FF0FF generates the curve when 0, +, and  are interpreted as turns and F as a unit stroke in the current direction.
%C Three copies of the curve can be arranged to create a reptile that is a lattice tiling, see the files "Tileplus" (axiom F+F+F), "Tileminus" (Axiom FFF), "Tilingplus" (selfsimilarity of the Tileplus), and "Complex numeration system" (giving the generalized unit square of a numeration system with base 1 + i * sqrt(12) that reproduces the Tileplus).
%H Joerg Arndt, <a href="/A265671/b265671.txt">Table of n, a(n) for n = 1..2197</a>
%H Joerg Arndt, <a href="/A265671/a265671.pdf">First iterate of the curve</a>, <a href="/A265671/a265671_1.pdf">Second iterate</a>, <a href="/A265671/a265671_2.pdf">Third iterate</a>, <a href="/A265671/a265671_3.pdf">Fourth iterate</a>.
%H Joerg Arndt, <a href="/A265671/a265671.png">Rendering used for the Tshirt on Neil's 75th birthday</a> (png image, 1716 X 2732 pixel).
%H Joerg Arndt, <a href="/A265671/a265671_4.pdf">Tileplus</a>, <a href="/A265671/a265671_5.pdf">Tileminus</a>, <a href="/A265671/a265671_6.pdf">Tilingplus</a>, <a href="/A265671/a265671_1.png">Complex numeration system</a>.
%H Joerg Arndt, <a href="http://arxiv.org/abs/1607.02433">Planefilling curves on all uniform grids</a>, arXiv:1607.02433 [math.CO], (8July2016)
%H <a href="/index/Ar#13automatic">Index entries for 13automatic sequences</a>.
%Y Cf. A029883, A035263, A060236, A080846, A156595, A175337, A176405, A176416.
%Y Cf. A234434 (curves on the triangular grid).
%Y Cf. A229214 (a similar Lsystem for Gosper's flowsnake).
%K nonn
%O 1,2
%A _Joerg Arndt_, Dec 13 2015
