login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

a(n) is the X-coordinate of the n-th point of the Kochawave curve; sequence A335381 gives Y-coordinates.
2

%I #22 Nov 01 2022 03:13:45

%S 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,

%T 16,16,17,18,18,19,18,18,19,22,20,21,20,18,19,18,18,19,18,18,19,20,20,

%U 21,22,22,23,24,24,25,24,24,25,26,26,27,28,28,29,30,30

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

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

%C Y

%C /

%C /

%C 0 ---- X

%C 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 .+ C

%C .../

%C ... /

%C ... /

%C +------>+. +------>+

%C A B D E

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

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

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

%C The distance between two consecutive points is related to A160381:

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

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

%H Rémy Sigrist, <a href="/A335380/b335380.txt">Table of n, a(n) for n = 0..4096</a>

%H Rémy Sigrist, <a href="https://arxiv.org/abs/2210.17320">The Kochawave curve, a variant of the Koch curve</a>, arXiv:2210.17320 [math.HO], 2022.

%H Rémy Sigrist, <a href="/A335380/a335380_1.png">Representation of the Kochawave curve</a>

%H Rémy Sigrist, <a href="/A335380/a335380.png">Representation of the first iterations of the Kochawave curve</a>

%H Rémy Sigrist, <a href="/A335380/a335380.gp.txt">PARI program for A335380</a>

%H <a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>

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

%e The Kochawave curve starts (on a hexagonal lattice) as follows:

%e . . . . . . + . . .

%e /|6

%e / |

%e / |

%e . . . . . . | . .+ . .

%e / | .../ 8

%e / | ... /

%e / | ... /

%e . . . . . . +. + . .

%e / 7 |9

%e / |

%e / |

%e . . .+ . .+ . +11 | . .+ .

%e .../ 2 ... 5 / \ | .../ 14

%e ... / ... / \ | ... /

%e ... / ... / \| ... /

%e +-------+. +-------+. . . +-------+. +-------+

%e 0 1 3 4 12 13 10 15 16

%e - hence a(8) = a(9) = a(11) = a(12) = 6.

%o (PARI) See Links section.

%Y Cf. A160381, A335381.

%K nonn

%O 0,3

%A _Rémy Sigrist_, Jun 04 2020