A341163 a(n) is the X-coordinate of the n-th point of the space filling curve A defined in Comments section; A341164 gives Y-coordinates.

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

Offset

0,3

Comments

Coordinates are given on a hexagonal lattice with X-axis and Y-axis as follows:

Y

/

/

0 ---- X

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

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

. __+__ .

__---- ----__

+ . . +

0

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

+

/B\

+ / \

/B\ /A C\

/ \ --> +-------+

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

+-------+ / \ / \

O /A C\A/B C\

+-------+-------+

O

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

This sequence has similarities with A341018.

Links

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

Zbigniew Fiedorowicz, The Peano Curve Theorem

Rémy Sigrist, Illustration of A_5

Rémy Sigrist, Illustration of the first bisection of A

Rémy Sigrist, Illustration of the first quadrisection of A

Rémy Sigrist, Illustration of the fourth quadrisection of A

Rémy Sigrist, PARI program for A341163

Index entries for sequences related to coordinates of 2D curves

Example

The curve A starts as follows:
                  .
                .   .
              .   5   .
            4   .   .   6
          .   .   3   .   .
        .   1   .   .   7   .
      0   .   .   2   .   .   8
- so a(0) = a(4) = 0,
     a(1) = a(5) = 1,
     a(3) = 2,
     a(2) = a(6) = 3,
     a(8) = 6.

Prog

(PARI) See Links section.

Crossrefs

Cf. A341018, A341164.

Sequence in context: A079618 A151844 A286223 * A329278 A348966 A008783

Adjacent sequences: A341160 A341161 A341162 * A341164 A341165 A341166

Keyword

nonn,look

Author

Rémy Sigrist, Feb 06 2021

Status

approved