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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A334476 a(n) is the Y-coordinate of the n-th point of the space filling curve T defined in A334474; sequence A334475 gives X-coordinates.

Original entry on oeis.org

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

Views

Author

Rémy Sigrist, May 02 2020

Keywords

Examples

			The space filling curve T starts as follows:
    5|  ...
     |   |
     |   |
    4|  16
     |   \
     |    \
    3|   5 15-14-13
     |   |\       |
     |   | \      |
    2|   4  6 11-12
     |    \  \  \
     |     \  \  \
    1|   1  3  7 10
     |   |\ |  |  |
     |   | \|  |  |
    0|   0  2  8--9
  ---+-------------
  y/x|   0  1  2  3
- hence a(1) = a(3) = a(7) = a(10) = 1.

Links

Crossrefs

Cf. A334474, A334475.

Programs

  • PARI
    See Links section.

Formula

A334474(a(n), A334475(n)) = n.

A342224 The n-th and a(n)-th points of the curve (A334474, A334475) are symmetrical with respect to the line X=Y.

Original entry on oeis.org

0, 2, 1, 3, 8, 9, 7, 6, 4, 5, 15, 11, 14, 13, 12, 10, 31, 32, 30, 29, 33, 35, 34, 28, 27, 26, 25, 24, 23, 19, 18, 16, 17, 20, 22, 21, 63, 62, 59, 58, 60, 61, 42, 44, 43, 47, 46, 45, 57, 55, 56, 51, 54, 53, 52, 49, 50, 48, 39, 38, 40, 41, 37, 36, 121, 122, 120
Offset: 0

Views

Author

Rémy Sigrist, Mar 06 2021

Keywords

Comments

In other words, a(n) is the unique k such that A334474(n) = A334475(k) and A334474(k) = A334475(n).
This sequence is a self-inverse permutation of the nonnegative integers.

Examples

			The curve (A334474, A334475) begins as follows on a hexagonal lattice:
                 +
                /5\
               /   \
              +4    +6
               \     \
                \     \
           +    3+     +7
          /1\   /     /
         /   \ /     /
        +     +     +----+
         0     2     8    9
- so a(0) = 0,
     a(1) = 2,
     a(3) = 3,
     a(4) = 8,
     a(5) = 9,
     a(6) = 7.

Links

Crossrefs

See A342217 and A342218 for similar sequences.
Cf. A007582, A334474, A334475.

Programs

  • PARI
    See Links section.

Formula

a(n) < A007582(k) for any n < A007582(k).

A334474 T(n, k) is the number of steps from the point (0, 0) to the point (k, n) along the space filling curve T defined in Comments section; square array T(n, k), n, k >= 0 read by antidiagonals downwards.

Original entry on oeis.org

0, 2, 1, 8, 3, 4, 9, 7, 6, 5, 31, 10, 11, 15, 16, 32, 30, 12, 14, 18, 17, 33, 29, 27, 13, 24, 19, 20, 35, 34, 28, 26, 25, 23, 22, 21, 121, 36, 37, 41, 42, 61, 62, 63, 64, 122, 120, 38, 40, 43, 44, 60, 59, 66, 65, 123, 119, 117, 39, 47, 46, 45, 58, 72, 67, 68
Offset: 0

Views

Author

Rémy Sigrist, May 02 2020

Keywords

Comments

We consider a hexagonal lattice with X-axis and Y-axis as follows:
Y
/
/
0 ---- X
We define the family {T_n, n > 0} as follows:
- T_1 contains the origin (0, 0):
+
O
- T_2 contains the points (0, 0), (0, 1) and (1, 0), in that order:
+
/ \
/ \
+ . . +
O
- for n > 1, T_{2*n} is built from 3 copies of T_n and one copy T_{n-1} arranged as follows:
+
/ \
/ \
/ n \
/ \
+ . . . . +
\ \
+ +-----+ +
/ \ .n-1/ / .
/ \ . / / .
/ n \ + / n .
/ \ / / .
+ . . . . + +---------+
O
- for n > 0, T_{2*n+1} is built from 3 copies of T_n and one copy of T_{n+1} arranged as follows:
+
/ \
/ \
/ \
/ n+1 \
/ \
+ . . . . . +
\ \
+ +---------+ +
/ \ . / / .
/ \ . n / / .
/ n \ . / / n .
/ \ . / / .
+ . . . . +--+ +---------+
O
- for any n > 0, T_n starts at (0, 0) and ends at (n-1, n-1), and contains every point (x, y) such that x, y >= 0 and x + y < n,
- T is the limit of T_{2^k} as k tends to infinity (note that for any k >= 0, T_{2^k} is a prefix of T_{2^(k+1)}),
- T visits exactly once every point (x, y) such that x, y >= 0.

Examples

			Square array starts:
  n\k|   0  1  2  3  4  5  6  7
  ---+-------------------------
    0|   0  2  8--9 31-32-33 35
     |   | /|  |  |  |     | /|
     |   |/ |  |  |  |     |/ |
    1|   1  3  7 10 30-29 34 36
     |     /  /  /      |    /
     |    /  /  /       |   /
    2|   4  6 11-12 27-28 37-38
     |   | /      |  |        |
     |   |/       |  |        |
    3|   5 15-14-13 26 41-40-39
     |    /         /  /
     |   /         /  /
    4|  16 18 24-25 42-43 47-48
     |   | /|  |       /  /   |
     |   |/ |  |      /  /    |
    5|  17 19 23 61 44 46 50-49
     |     /  /  /|  | /   |
     |    /  /  / |  |/    |
    6|  20 22 62 60 45 56 51-52-
     |   | /  /  /     /|

Links

Crossrefs

See A334475 and A334476 for the coordinates of the curve.

Programs

  • PARI
    \\ See Links section.

Formula

T(A334476(n), A334475(n)) = n.
Showing 1-3 of 3 results.

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