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

%I A349198 #12 Dec 30 2024 01:20:33
%S A349198 0,0,1,1,0,0,-1,-1,0,0,1,0,1,1,2,2,3,2,3,3,4,4,3,3,2,2,3,3,2,2,1,2,1,
%T A349198 2,1,1,0,0,1,1,0,0,-1,-1,0,0,-1,-1,-2,-1,-2,-1,-2,-2,-3,-3,-2,-2,-3,
%U A349198 -3,-4,-4,-3,-3,-2,-3,-2,-2,-1,-1,0,-1,0,0,1,1,0
%N A349198 a(n) is the Y-coordinate of the n-th point of the alternate terdragon curve; sequence A349197 gives X-coordinates.
%C A349198 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 A349198            Y
%C A349198           /
%C A349198          /
%C A349198         0 ---- X
%C A349198 The alternate terdragon curve can be represented using an L-system.
%H A349198 Rémy Sigrist, <a href="/A349198/b349198.txt">Table of n, a(n) for n = 0..6561</a>
%H A349198 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.
%H A349198 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 A349198 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 A349198 Rémy Sigrist, <a href="/A349198/a349198.gp.txt">PARI program for A349198</a>
%H A349198 <a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>
%F A349198 a(9^k) = 0 for any k >= 0.
%F A349198 a(9*n) = 3*a(n).
%e A349198 The alternate terdragon curve starts as follows:
%e A349198                   14
%e A349198                    \
%e A349198                     \
%e A349198          2----3,12--10,13
%e A349198           \   / \   / \
%e A349198            \ /   \ /   \
%e A349198       0----1,4--5,8,11--9
%e A349198                  / \
%e A349198                 /   \
%e A349198                6-----7
%e A349198 - so a(0) = a(1) = a(4) = a(5) = a(8) = a(9) = a(11) = 0,
%e A349198      a(6) = a(7) = -1,
%e A349198      a(2) = a(3) = a(10) = a(12) = a(13) = 1.
%o A349198 (PARI) See Links section.
%Y A349198 See A349041 for a similar sequence.
%Y A349198 Cf. A349197.
%K A349198 sign
%O A349198 0,15
%A A349198 _Rémy Sigrist_, Nov 10 2021