A341164 -id:A341164 - cp's OEIS frontend

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

Rémy Sigrist, Feb 06 2021

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.

			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.

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

Cf. A341018, A341164.

PARI
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cp's OEIS Frontend

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.

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