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

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, 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, -3, -4, -4, -3, -3, -2, -3, -2, -2, -1, -1, 0, -1, 0, 0, 1, 1, 0

Offset: 0

Author

Rémy Sigrist, Nov 10 2021

Keywords

• sign

Comments

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):

Y

/

/

0 ---- X

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

Examples

			The alternate terdragon curve starts as follows:
                  14
                   \
                    \
         2----3,12--10,13
          \   / \   / \
           \ /   \ /   \
      0----1,4--5,8,11--9
                 / \
                /   \
               6-----7
- so a(0) = a(1) = a(4) = a(5) = a(8) = a(9) = a(11) = 0,
     a(6) = a(7) = -1,
     a(2) = a(3) = a(10) = a(12) = a(13) = 1.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..6561
• 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, Selected Papers on Fun and Games, 2011, pages 571-614.
• Chandler Davis and Donald E. Knuth, Number Representations and Dragon Curves, 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]
• Rémy Sigrist, Colored representation of the first 1 + 9^6 points of the alternate terdragon curve (where the hue is function of the number of steps from the origin)
• Rémy Sigrist, PARI program for A349198
• Index entries for sequences related to coordinates of 2D curves

Crossrefs

See A349041 for a similar sequence.

Cf. A349197.

Programs

PARI

See Links section.

Formula

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

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