A335380 a(n) is the X-coordinate of the n-th point of the Kochawave curve; sequence A335381 gives Y-coordinates.

0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 6, 6, 7, 8, 8, 9, 10, 10, 11, 12, 12, 9, 12, 12, 13, 16, 14, 15, 16, 16, 17, 18, 18, 19, 18, 18, 19, 22, 20, 21, 20, 18, 19, 18, 18, 19, 18, 18, 19, 20, 20, 21, 22, 22, 23, 24, 24, 25, 24, 24, 25, 26, 26, 27, 28, 28, 29, 30, 30

Offset: 0

Author

Rémy Sigrist, Jun 04 2020

Keywords

• nonn

Comments

Coordinates are given on a hexagonal lattice with X-axis and Y-axis as follows:

Y

/

/

0 ---- X

The Kochawave curve is a variant of the Koch curve that can be built by successively applying the following substitution to an initial vector (1, 0):

.+ C

.../

... /

... /

+------>+. +------>+

A B D E

- the points A, B, D and E are aligned and equally spaced,

- the points D, C and E form an equilateral triangle

(for the Koch curve, the points B, C and D form an equilateral triangle).

The distance between two consecutive points is related to A160381:

- for any n >= 0, let z(n) = a(n) + A335381(n) * exp(i*Pi/3) (where i denotes the imaginary unit),

- the square of the distance from z(n) to z(n+1) is 3^A160381(n).

Examples

			The Kochawave curve starts (on a hexagonal lattice) as follows:
    .       .       .       .       .       .       +       .       .       .
                                                   /|6
                                                  / |
                                                 /  |
        .       .       .       .       .       .   |   .      .+       .       .
                                               /    |       .../ 8
                                              /     |    ...  /
                                             /      | ...    /
    .       .       .       .       .       .       +.      +       .       .
                                           /         7      |9
                                          /                 |
                                         /                  |
        .       .      .+       .      .+       .       +11 |   .      .+       .
                    .../ 2          ...  5             / \  |       .../ 14
                 ...  /          ...                  /   \ |    ...  /
              ...    /        ...                    /     \| ...    /
    +-------+.      +-------+.      .       .       +-------+.      +-------+
     0       1       3       4                       12   13 10      15      16
- hence a(8) = a(9) = a(11) = a(12) = 6.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..4096
• Rémy Sigrist, The Kochawave curve, a variant of the Koch curve, arXiv:2210.17320 [math.HO], 2022.
• Rémy Sigrist, Representation of the Kochawave curve
• Rémy Sigrist, Representation of the first iterations of the Kochawave curve
• Rémy Sigrist, PARI program for A335380
• Index entries for sequences related to coordinates of 2D curves

Crossrefs

Cf. A160381, A335381.

Programs

PARI

See Links section.

Formula

a(4^k) = 3^k for any k >= 0.