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

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, -3, -4, -3, -4, -4, -5, -5, -4, -4, -5, -5, -6, -6, -5, -5, -4, -5, -4, -4, -3, -3, -4, -4, -5, -5, -4, -4, -5, -5, -6, -5, -6, -5, -6, -6, -7, -6, -7, -6, -7, -7, -8

Offset: 0

Author

Rémy Sigrist, Nov 06 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 terdragon curve can be represented using an L-system.

A062756, when interpreted as a sequence of directions A062756(n)*120 degrees, yields the same curve.

Examples

			The terdragon curve starts (on a hexagonal lattice) as follows:
              +-----+
              8\    9
                \
           +-----+7
           6\   /4\
             \5/   \
              +-----+
              2\   3
                \
           +-----+
           0     1
- so a(0) = a(2) = a(4) = a(5) = a(7) = a(9) = 0,
     a(1) = a(3) = 1,
     a(6) = a(8) = -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. See section 5 delta(n) for zeta = third root of unity.
• 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]
• Kevin Ryde, Iterations of the Terdragon Curve, see index "point".
• Rémy Sigrist, Colored representation of the first 1 + 3^11 points of the terdragon curve (where the hue is function of the number of steps from the origin)
• Rémy Sigrist, PARI program for A349040
• Wikipedia, Terdragon
• Index entries for sequences related to coordinates of 2D curves

Crossrefs

Cf. A080846 (turn), A062756 (segment direction), A349041.

Programs

PARI

See Links section.