A341120 - OEIS

%I #26 Mar 07 2021 10:49:27

%S 0,0,1,2,2,3,3,3,2,3,3,3,2,2,1,0,0,1,1,1,0,0,1,2,2,2,3,4,4,3,3,3,4,5,

%T 5,5,4,4,5,6,6,6,7,8,8,7,7,7,8,8,7,6,6,5,5,5,6,5,5,5,6,6,7,8,8,9,9,9,

%U 8,8,9,10,10,10,11,12,12,11,11,11,12,12,13

%N a(n) is the X-coordinate of the n-th point of the space filling curve C defined in Comments section; A341121 gives Y-coordinates.

%C We define the family {C_k, k >= 0}, as follows:

%C - C_0 corresponds to the points (0, 0), (0, 1), (1, 1), (2, 1) and (2, 0), in that order:

%C        +---+---+

%C        |       |

%C        +       +

%C       O

%C - for any k >= 0, C_{k+1} is obtained by arranging 4 copies of C_k as follows:

%C                            + . . . + . . . +

%C                            .   B   .   B   .

%C        + . . . +           .       .       .

%C        .   B   .           .A     C.A     C.

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

%C        .A     C.           .C      .      A.

%C        + . . . +           .      B.B      .

%C       O                    .A      .      C.

%C                            + . . . + . . . +

%C                           O

%C - the space filling curve C is the limit of C_{2*k} as k tends to infinity.

%C The even bisection of the curve M defined in A341018 is similar to C and vice versa.

%C The third quadrisection of C is similar to the Hilbert Hamiltonian walk H = A059252, A059253.

%C H is the number of points in the middle of each unit square in Hilbert's subdivisions, whereas here points are at the starting corner of each unit square. This start is either the bottom left or top right corner depending on how many 180-degree rotations have been applied. These rotations are digit 3's of n written in base 4, hence the formula below adding A283316.

%H Rémy Sigrist, <a href="/A341120/b341120.txt">Table of n, a(n) for n = 0..16384</a>

%H F. M. Dekking, <a href="http://dx.doi.org/10.1016/0001-8708(82)90066-4">Recurrent Sets</a>, Advances in Mathematics, vol. 44, no. 1, 1982. See section 4.8.

%H David Hilbert, <a href="https://doi.org/10.1007/BF01199431">Ueber die stetige Abbildung einer Linie auf ein Flächenstück</a>, Mathematische Annalen, volume 38, number 3, 1891, pages 459-460. Also <a href="https://eudml.org/doc/157555">EUDML</a> (link to GDZ).

%H Rémy Sigrist, <a href="/A341120/a341120.png">Illustration of C_6</a>

%H Rémy Sigrist, <a href="/A341120/a341120_1.png">Illustration of the connection of C with the Hilbert Hamiltonian walk</a>

%H Rémy Sigrist, <a href="/A341120/a341120_2.png">Illustration of the connection of C with the space filling curve M defined in A341018</a>

%H Rémy Sigrist, <a href="/A341120/a341120.gp.txt">PARI program for A341120</a>

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

%F a(n) = A341121(n) - A059285(n).

%F a(n) = A341121(n) iff n belongs to A062880.

%F a(2*n) = A341018(n).

%F a(4*n) = 2*A341121(n).

%F a(16*n) = 4*a(n).

%F a(n) = A059252(n) + A283316(n+1).

%F A059253(n) = (a(4*n+2)-1)/2.

%e Points n and their locations X=a(n), Y=A341121(n) begin as follows. n=7 and n=9 are both at X=3,Y=2, and n=11,n=31 both at X=3,Y=4.

%e       |       |

%e     4 | 16---17   12--11,31

%e       |  |         |    |

%e     3 | 15---14---13   10

%e       |                 |

%e     2 |            8---7,9

%e       |                 |

%e     1 |  1----2----3    6

%e       |  |         |    |

%e   Y=0 |  0         4----5

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

%e        X=0    1    2    3

%o (PARI) See Links section.

%Y Cf. A341121 (Y coordinate), A059285 (projection Y-X), A062880 (n on X=Y diagonal).

%Y Cf, A059252, A341018.

%K nonn

%O 0,4

%A _Kevin Ryde_ and _Rémy Sigrist_, Feb 05 2021