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

0,13

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.

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.

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

Example

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.

Prog

(PARI) See Links section.

Crossrefs

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

Sequence in context: A153864 A345128 A112399 * A165123 A318439 A106180

Adjacent sequences: A349037 A349038 A349039 * A349041 A349042 A349043

Keyword

sign

Author

Rémy Sigrist, Nov 06 2021

Status

approved