A349041 -id:A349041 - cp's OEIS frontend

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

Rémy Sigrist, Nov 06 2021

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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.

			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.

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

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

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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.

Original entry on oeis.org

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

Rémy Sigrist, Nov 10 2021

sign

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.

			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.

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

See A349041 for a similar sequence.

Cf. A349197.

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

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

A349219 a(n) is the Y-coordinate of the n-th point of the 7-dragon curve; sequence A349218 gives X-coordinates.

Original entry on oeis.org

0, 0, 1, 1, 0, 0, 1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4, 4, 3, 3, 4, 4, 3, 3, 2, 3, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 3, 2, 3, 3, 4, 3, 4, 4, 5, 5, 4, 4, 5, 5, 6, 5, 6, 6, 7, 6, 7, 6, 6, 5, 6, 5, 5, 4, 5, 4, 5, 5, 6, 5, 6, 6, 7, 7, 6, 6, 7, 7, 8, 7, 8, 8, 9, 8, 9, 8, 8

Offset: 0

Rémy Sigrist, Nov 11 2021

sign

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 7-dragon curve can be represented using an L-system.

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

Rémy Sigrist, Table of n, a(n) for n = 0..16807
Rémy Sigrist, Colored representation of the first 1 + 7^7 points of the 7-dragon curve (where the hue is function of the number of steps from the origin)
Rémy Sigrist, PARI program for A349219
Jeffrey Ventrella, Brainfilling Curves: The Root 7 Family
Index entries for sequences related to coordinates of 2D curves

See A349041 and A349198 for similar sequences.

Cf. A349218.

PARI
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See Links section.
```

A349320 a(n) is the Y-coordinate of the n-th point of the hexdragon curve; sequence A349319 gives X-coordinates.

Original entry on oeis.org

0, 0, 1, 2, 3, 3, 4, 5, 5, 4, 4, 5, 6, 6, 7, 8, 9, 9, 10, 11, 11, 10, 10, 11, 11, 10, 9, 9, 8, 7, 7, 8, 8, 7, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 14, 13, 13, 14, 15, 15, 16, 17, 18, 18, 19, 20, 20, 19, 19, 20, 20, 19, 18, 18, 17, 16, 16, 17, 17, 16, 16, 17, 17

Offset: 0

Rémy Sigrist, Nov 14 2021

sign

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 hexdragon curve can be represented using an L-system obtained from that of the terdragon curve by replacing each "move forward and turn +-120 degrees" step by two "move forward and turn +- 60 degrees" steps.

			The hexdragon curve starts as follows:
              16-17
              /
            15
              \
              14
              /
         12-13
         /
       11      8--7
         \    /    \
         10--9      6
                   /
               4--5
              /
             3
              \
               2
              /
          0--1
- so a(0) = a(1) = 0,
     a(2) = 1,
     a(3) = 2,
     a(4) = a(5) = 3,
     a(6) = a(9) = a(10) = 4,
     a(7) = a(8) = a(11) = 5,
     a(12) = a(13) = 6,
     a(14) = 7,
     a(15) = 8,
     a(16) = a(17) = 9.

Rémy Sigrist, Table of n, a(n) for n = 0..4373
Joerg Arndt, Matters Computational (The Fxtbook), section 1.31.4 Terdragon and hexdragon.
Rémy Sigrist, Representation of the hexdragon curve after 6 iterations
Rémy Sigrist, PARI program for A349320
Index entries for sequences related to coordinates of 2D curves

See A349041 for a similar sequence.

Cf. A349319.

PARI
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See Links section.
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cp's OEIS Frontend

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

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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.

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A349219 a(n) is the Y-coordinate of the n-th point of the 7-dragon curve; sequence A349218 gives X-coordinates.

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PARI

A349320 a(n) is the Y-coordinate of the n-th point of the hexdragon curve; sequence A349319 gives X-coordinates.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI