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A349197 a(n) is the X-coordinate of the n-th point of the alternate terdragon curve; sequence A349198 gives Y-coordinates.

%I A349197 #15 Dec 30 2024 01:20:02
%S A349197 0,1,0,1,1,2,2,3,2,3,2,2,1,2,1,2,1,1,0,1,0,1,1,2,2,3,2,3,3,4,4,3,3,2,
%T A349197 2,3,3,4,3,4,4,5,5,6,5,6,6,7,7,6,6,5,5,6,6,7,6,7,7,8,8,9,8,9,8,8,7,8,
%U A349197 7,8,7,7,6,7,6,7,7,8,8,9,8,9,8,8,7,8,7
%N A349197 a(n) is the X-coordinate of the n-th point of the alternate terdragon curve; sequence A349198 gives Y-coordinates.
%C A349197 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 A349197            Y
%C A349197           /
%C A349197          /
%C A349197         0 ---- X
%C A349197 The alternate terdragon curve can be represented using an L-system.
%H A349197 Rémy Sigrist, <a href="/A349197/b349197.txt">Table of n, a(n) for n = 0..6561</a>
%H A349197 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 end of section 5.
%H A349197 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 A349197 Kevin Ryde, <a href="http://user42.tuxfamily.org/terdragon/index.html">Iterations of the Terdragon Curve</a>, see index "AltPoint".
%H A349197 Rémy Sigrist, <a href="/A349197/a349197.png">Colored representation of the first 1 + 9^6 points of the alternate terdragon curve</a> (where the hue is function of the number of steps from the origin)
%H A349197 Rémy Sigrist, <a href="/A349197/a349197.gp.txt">PARI program for A349197</a>
%H A349197 <a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>
%F A349197 a(9^k) = 3^k for any k >= 0.
%F A349197 a(9*n) = 3*a(n).
%e A349197 The alternate terdragon curve starts as follows:
%e A349197                   14
%e A349197                    \
%e A349197                     \
%e A349197          2----3,12--10,13
%e A349197           \   / \   / \
%e A349197            \ /   \ /   \
%e A349197       0----1,4--5,8,11--9
%e A349197                  / \
%e A349197                 /   \
%e A349197                6-----7
%e A349197 - so a(0) = a(2) = 0,
%e A349197      a(1) = a(3) = a(4) = a(12) = a(14) = 1.
%o A349197 (PARI) See Links section.
%Y A349197 See A349040 for a similar sequence.
%Y A349197 Cf. A349198.
%K A349197 nonn
%O A349197 0,6
%A A349197 _Rémy Sigrist_, Nov 10 2021