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

%I A335380 #22 Nov 01 2022 03:13:45
%S A335380 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 A335380 16,16,17,18,18,19,18,18,19,22,20,21,20,18,19,18,18,19,18,18,19,20,20,
%U A335380 21,22,22,23,24,24,25,24,24,25,26,26,27,28,28,29,30,30
%N A335380 a(n) is the X-coordinate of the n-th point of the Kochawave curve; sequence A335381 gives Y-coordinates.
%C A335380 Coordinates are given on a hexagonal lattice with X-axis and Y-axis as follows:
%C A335380            Y
%C A335380           /
%C A335380          /
%C A335380         0 ---- X
%C A335380 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 A335380                            .+ C
%C A335380                         .../
%C A335380                      ...  /
%C A335380                   ...    /
%C A335380         +------>+.      +------>+
%C A335380         A       B       D       E
%C A335380 - the points A, B, D and E are aligned and equally spaced,
%C A335380 - the points D, C and E form an equilateral triangle
%C A335380   (for the Koch curve, the points B, C and D form an equilateral triangle).
%C A335380 The distance between two consecutive points is related to A160381:
%C A335380 - for any n >= 0, let z(n) = a(n) + A335381(n) * exp(i*Pi/3) (where i denotes the imaginary unit),
%C A335380 - the square of the distance from z(n) to z(n+1) is 3^A160381(n).
%H A335380 Rémy Sigrist, <a href="/A335380/b335380.txt">Table of n, a(n) for n = 0..4096</a>
%H A335380 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 A335380 Rémy Sigrist, <a href="/A335380/a335380_1.png">Representation of the Kochawave curve</a>
%H A335380 Rémy Sigrist, <a href="/A335380/a335380.png">Representation of the first iterations of the Kochawave curve</a>
%H A335380 Rémy Sigrist, <a href="/A335380/a335380.gp.txt">PARI program for A335380</a>
%H A335380 <a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>
%F A335380 a(4^k) = 3^k for any k >= 0.
%e A335380 The Kochawave curve starts (on a hexagonal lattice) as follows:
%e A335380     .       .       .       .       .       .       +       .       .       .
%e A335380                                                    /|6
%e A335380                                                   / |
%e A335380                                                  /  |
%e A335380         .       .       .       .       .       .   |   .      .+       .       .
%e A335380                                                /    |       .../ 8
%e A335380                                               /     |    ...  /
%e A335380                                              /      | ...    /
%e A335380     .       .       .       .       .       .       +.      +       .       .
%e A335380                                            /         7      |9
%e A335380                                           /                 |
%e A335380                                          /                  |
%e A335380         .       .      .+       .      .+       .       +11 |   .      .+       .
%e A335380                     .../ 2          ...  5             / \  |       .../ 14
%e A335380                  ...  /          ...                  /   \ |    ...  /
%e A335380               ...    /        ...                    /     \| ...    /
%e A335380     +-------+.      +-------+.      .       .       +-------+.      +-------+
%e A335380      0       1       3       4                       12   13 10      15      16
%e A335380 - hence a(8) = a(9) = a(11) = a(12) = 6.
%o A335380 (PARI) See Links section.
%Y A335380 Cf. A160381, A335381.
%K A335380 nonn
%O A335380 0,3
%A A335380 _Rémy Sigrist_, Jun 04 2020