A341019 a(n) is the Y-coordinate of the n-th point of the space filling curve M defined in Comments section; A341018 gives X-coordinates.

0, 1, 0, 1, 2, 3, 4, 3, 4, 5, 6, 7, 6, 7, 6, 5, 4, 5, 6, 7, 6, 7, 6, 5, 4, 3, 4, 3, 2, 1, 0, 1, 0, 1, 2, 3, 2, 3, 2, 1, 0, 1, 0, 1, 2, 3, 4, 3, 4, 5, 4, 5, 6, 7, 8, 7, 8, 7, 6, 5, 6, 5, 6, 7, 8, 9, 10, 11, 10, 11, 10, 9, 8, 9, 8, 9, 10, 11, 12, 11, 12, 13, 12

Offset

0,5

Comments

We define the family {M_n, n >= 0}, as follows:

- M_0 corresponds to the points (0, 0), (1, 1) and (2, 0), in that order:

+

/ \

/ \

+ +

O

- for any n >= 0, M_{n+1} is obtained by arranging 4 copies of M_n as follows:

+ . . . + . . . +

. B . B .

+ . . . + . . .

. B . .A C.A C.

. . --> + . . . + . . . +

.A C. .C . A.

+ . . . + . B.B .

O .A . C.

+ . . . + . . . +

O

- for any n >= 0, M_n has A087289(n) points,

- the space filling curve M is the limit of M_{2*n} as n tends to infinity.

The odd bisection of M is similar to a Hilbert's Hamiltonian walk (hence the connection with A059252).

Links

Rémy Sigrist, Table of n, a(n) for n = 0..8192

F. M. Dekking, Recurrent Sets, Advances in Mathematics, vol. 44, no. 1, 1982.

Larry Riddle, Space Filling Curve

Rémy Sigrist, PARI program for A341019

Index entries for sequences related to coordinates of 2D curves

Formula

A059252(n) = (a(2*n+1)-1)/2.

a(4*n) = 2*A341018(n).

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

Example

The curve M starts as follows:
       11+ 13+   +19 +21
        / \ / \ / \ / \
     10+ 12+ 14+18 +20 +22
        \     / \     /
        9+ 15+   +17 +23
        /     \ /     \
      8+  6+   +   +26 +24
        \ / \ 16  / \ /
        7+  5+   +27 +25
            /     \
          4+       +28
            \     /
        1+  3+   +29 +31
        / \ /     \ / \
      0+  2+       +30 +32
- so a(0) = a(2) = a(30) = a(32) = 0,
     a(1) = a(3) = a(29) = a(31) = 1.

Prog

(PARI) See Links section.

Crossrefs

Cf. A059252, A087289, A341018.

Sequence in context: A325953 A245343 A369858 * A360535 A255938 A081748

Adjacent sequences: A341016 A341017 A341018 * A341020 A341021 A341022

Keyword

nonn

Author

Rémy Sigrist, Feb 02 2021

Status

approved