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A349040 a(n) is the X-coordinate of the n-th point of the terdragon curve; sequence A349041 gives Y-coordinates.

%I A349040 #22 Dec 30 2024 01:19:22
%S A349040 0,1,0,1,0,0,-1,0,-1,0,-1,-1,-2,-2,-1,-1,-2,-2,-3,-2,-3,-2,-3,-3,-4,
%T A349040 -3,-4,-3,-4,-4,-5,-5,-4,-4,-5,-5,-6,-6,-5,-5,-4,-5,-4,-4,-3,-3,-4,-4,
%U A349040 -5,-5,-4,-4,-5,-5,-6,-5,-6,-5,-6,-6,-7,-6,-7,-6,-7,-7,-8
%N A349040 a(n) is the X-coordinate of the n-th point of the terdragon curve; sequence A349041 gives Y-coordinates.
%C A349040 Coordinates are given on a hexagonal lattice with X-axis and Y-axis as follows (the Y-axis corresponds to the sixth primitive root of unity):
%C A349040            Y
%C A349040           /
%C A349040          /
%C A349040         0 ---- X
%C A349040 The terdragon curve can be represented using an L-system.
%C A349040 A062756, when interpreted as a sequence of directions A062756(n)*120 degrees, yields the same curve.
%H A349040 Rémy Sigrist, <a href="/A349040/b349040.txt">Table of n, a(n) for n = 0..6561</a>
%H A349040 Chandler Davis and Donald E. Knuth, Number Representations and Dragon Curves -- I and II, Journal of Recreational Mathematics, volume 3, number 2, April 1970, pages 66-81, and number 3, July 1970, pages 133-149. Reprinted in Donald E. Knuth, <a href="http://www-cs-faculty.stanford.edu/~uno/fg.html">Selected Papers on Fun and Games</a>, 2011, pages 571-614. See section 5 delta(n) for zeta = third root of unity.
%H A349040 Chandler Davis and Donald E. Knuth, <a href="/A005811/a005811.pdf">Number Representations and Dragon Curves</a>, Journal of Recreational Mathematics, volume 3, number 2, April 1970, pages 66-81, and number 3, July 1970, pages 133-149. [Cached copy, with permission]
%H A349040 Kevin Ryde, <a href="http://user42.tuxfamily.org/terdragon/index.html">Iterations of the Terdragon Curve</a>, see index "point".
%H A349040 Rémy Sigrist, <a href="/A349040/a349040.png">Colored representation of the first 1 + 3^11 points of the terdragon curve</a> (where the hue is function of the number of steps from the origin)
%H A349040 Rémy Sigrist, <a href="/A349040/a349040.gp.txt">PARI program for A349040</a>
%H A349040 Wikipedia, <a href="https://en.wikipedia.org/wiki/Dragon_curve#Terdragon">Terdragon</a>
%H A349040 <a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>
%e A349040 The terdragon curve starts (on a hexagonal lattice) as follows:
%e A349040               +-----+
%e A349040               8\    9
%e A349040                 \
%e A349040            +-----+7
%e A349040            6\   /4\
%e A349040              \5/   \
%e A349040               +-----+
%e A349040               2\   3
%e A349040                 \
%e A349040            +-----+
%e A349040            0     1
%e A349040 - so a(0) = a(2) = a(4) = a(5) = a(7) = a(9) = 0,
%e A349040      a(1) = a(3) = 1,
%e A349040      a(6) = a(8) = -1.
%o A349040 (PARI) See Links section.
%Y A349040 Cf. A080846 (turn), A062756 (segment direction), A349041.
%K A349040 sign
%O A349040 0,13
%A A349040 _Rémy Sigrist_, Nov 06 2021