OFFSET
0,3
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).
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, PARI program for A335380
FORMULA
a(4^k) = 3^k for any k >= 0.
EXAMPLE
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.
PROG
(PARI) See Links section.
CROSSREFS
KEYWORD
nonn
AUTHOR
Rémy Sigrist, Jun 04 2020
STATUS
approved