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a(n) is the X-coordinate of the n-th point of the alternate terdragon curve; sequence A349198 gives Y-coordinates.
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%I #15 Dec 30 2024 01:20:02

%S 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 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 7,8,7,7,6,7,6,7,7,8,8,9,8,9,8,8,7,8,7

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

%C 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 Y

%C /

%C /

%C 0 ---- X

%C The alternate terdragon curve can be represented using an L-system.

%H Rémy Sigrist, <a href="/A349197/b349197.txt">Table of n, a(n) for n = 0..6561</a>

%H 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 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 Kevin Ryde, <a href="http://user42.tuxfamily.org/terdragon/index.html">Iterations of the Terdragon Curve</a>, see index "AltPoint".

%H 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 Rémy Sigrist, <a href="/A349197/a349197.gp.txt">PARI program for A349197</a>

%H <a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>

%F a(9^k) = 3^k for any k >= 0.

%F a(9*n) = 3*a(n).

%e The alternate terdragon curve starts as follows:

%e 14

%e \

%e \

%e 2----3,12--10,13

%e \ / \ / \

%e \ / \ / \

%e 0----1,4--5,8,11--9

%e / \

%e / \

%e 6-----7

%e - so a(0) = a(2) = 0,

%e a(1) = a(3) = a(4) = a(12) = a(14) = 1.

%o (PARI) See Links section.

%Y See A349040 for a similar sequence.

%Y Cf. A349198.

%K nonn

%O 0,6

%A _Rémy Sigrist_, Nov 10 2021