

A265671


Directions of edges in a planefilling curve of order 13.


1



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
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OFFSET

1,2


COMMENTS

Infinite ternary word generated from the axiom 1 by the Lindenmayer system with maps 1 > 1222131123221, 2 > 2333212231332, and 3 > 3111323312113.
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}.
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".
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.
The shape of the curve is one of the A234434(13)=15 possible shapes.
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.
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).


LINKS

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 Tshirt on Neil's 75th birthday (png image, 1716 X 2732 pixel).
Joerg Arndt, Tileplus, Tileminus, Tilingplus, Complex numeration system.
Joerg Arndt, Planefilling curves on all uniform grids, arXiv:1607.02433 [math.CO], (8July2016)
Index entries for 13automatic sequences.


CROSSREFS

Cf. A029883, A035263, A060236, A080846, A156595, A175337, A176405, A176416.
Cf. A234434 (curves on the triangular grid).
Cf. A229214 (a similar Lsystem for Gosper's flowsnake).
Sequence in context: A308367 A205599 A327891 * A214707 A083039 A106253
Adjacent sequences: A265668 A265669 A265670 * A265672 A265673 A265674


KEYWORD

nonn


AUTHOR

Joerg Arndt, Dec 13 2015


STATUS

approved



