%I #14 Feb 09 2021 22:02:30

%S 0,1,0,2,3,4,3,1,0,2,3,4,3,5,6,5,6,7,6,8,9,10,9,7,6,4,3,2,3,1,0,1,0,2,

%T 3,4,3,5,6,5,6,7,6,8,9,10,9,7,6,8,9,10,9,11,12,11,12,11,12,10,9,8,9,

%U 11,12,13,12,14,15,16,15,13,12,14,15,16,15,17

%N a(n) is the Y-coordinate of the n-th point of the space filling curve A defined in Comments section; A341163 gives X-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 We define the family {A_n, n >= 0} as follows:

%C - A_0 corresponds to the points (0, 0), (1, 1) and (3, 0), in that order:

%C . __+__ .

%C __---- ----__

%C + . . +

%C 0

%C - for any n >= 0, A_{n+1} is obtained by arranging 4 copies of A_n as follows:

%C +

%C /B\

%C + / \

%C /B\ /A C\

%C / \ --> +-------+

%C /A C\ /B\C B/A\

%C +-------+ / \ / \

%C O /A C\A/B C\

%C +-------+-------+

%C O

%C - the space filling curve A is the limit of A_n as n tends to infinity.

%H Rémy Sigrist, <a href="/A341164/b341164.txt">Table of n, a(n) for n = 0..8192</a>

%H Zbigniew Fiedorowicz, <a href="https://people.math.osu.edu/fiedorowicz.1/math655/peano_t.html">The Peano Curve Theorem</a>

%H Rémy Sigrist, <a href="/A341164/a341164.gp.txt">PARI program for A341164</a>

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

%e The curve A starts as follows:

%e .

%e . .

%e . 5 .

%e 4 . . 6

%e . . 3 . .

%e . 1 . . 7 .

%e 0 . . 2 . . 8

%e - so a(0) = a(2) = a(8) = 0,

%e a(1) = a(7) = 1,

%e a(3) = 2,

%e a(4) = a(6) = 3,

%e a(5) = 4.

%o (PARI) See Links section.

%Y Cf. A341163.

%K nonn,look

%O 0,4

%A _Rémy Sigrist_, Feb 06 2021