A339455 - OEIS

%I #17 Feb 09 2021 11:00:50

%S 0,1,0,1,1,1,2,1,2,1,0,1,0,1,1,1,2,1,2,2,2,2,2,3,2,3,3,3,4,3,4,3,2,3,

%T 2,3,3,3,4,3,4,3,2,1,0,1,0,1,1,1,2,1,2,1,0,1,0,1,1,1,2,1,2,2,2,2,2,3,

%U 2,3,3,3,4,3,4,3,2,3,2,3,3,3,4,3,4,4,4

%N a(n) is the X-coordinate of the n-th point of the space filling curve H defined in Comments section; A339456 gives Y-coordinates.

%C We consider a hexagonal lattice with X-axis and Y-axis as follows:

%C            Y

%C           /

%C          /

%C         0 ---- X

%C We define the family {H_n, n > 0} as follows:

%C - T_1 contains the origin (0, 0) and (1, 0), in that order:

%C           +-->--+

%C          O

%C - for any n > 0, H_{n+1} is built from 4 copies of H_n connected with 2^(n+1) unit segments as follows:

%C                +->-2->-+

%C                 \     /

%C                  ^   v

%C                   \ /

%C            +->-1->-+->-4->-+

%C           O       / \

%C                  v   ^

%C                 /     \

%C                +->-3->-+

%C - H is the limit of H_n as n tends to infinity,

%C - H visits once every unit segment (u, v) where u and v are lattice points and at least one of u or v belongs to the region { (x, y) | x > 0 or x + y > 0 },

%C - the n-th segment of curve H has length 2^A235127(n).

%H Rémy Sigrist, <a href="/A339455/b339455.txt">Table of n, a(n) for n = 0..3008</a>

%H Rémy Sigrist, <a href="/A339455/a339455.png">Illustration of H_3</a>

%H Rémy Sigrist, <a href="/A339455/a339455.gp.txt">PARI program for A339455</a>

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

%o (PARI) See Links section.

%Y Cf. A235127, A339456.

%K nonn

%O 0,7

%A _Rémy Sigrist_, Dec 06 2020