A163357 -id:A163357 - cp's OEIS frontend

A163361 Hilbert curve in N x N grid, one-based, starting rightwards from the top-left corner.

1, 2, 4, 15, 3, 5, 16, 14, 8, 6, 17, 13, 9, 7, 59, 20, 18, 12, 10, 58, 60, 21, 19, 31, 11, 55, 57, 61, 22, 24, 30, 32, 54, 56, 62, 64, 235, 23, 25, 29, 33, 53, 51, 63, 65, 236, 234, 26, 28, 36, 34, 52, 50, 68, 66, 237, 233, 231, 27, 37, 35, 47, 49, 69, 67, 79, 240, 238, 232

Offset: 1

Antti Karttunen, Jul 29 2009

Inverse: A163362. Transpose: A163363.

a(n) = A163357(n-1)+1

A165467 Positions of zeros in A165466. Fixed points of A166043/A166044.

Original entry on oeis.org

0, 8, 9, 105, 1126, 6643718, 6643719, 6643727, 6643728, 6643729, 6643735, 6643736, 6643743, 6643744, 6643745, 6643752, 7746856, 7746857, 7746886, 7746887, 7746888, 7746889, 7747606, 7747718, 7747719, 7747720, 7747737

Offset: 0

Antti Karttunen, Oct 06 2009

nonn

Here is a little parable for illustrating the magnitudes of the numbers involved. Consider two immortal sage kings traveling on the infinite chessboard, visiting every square at a leisurely pace of one square per day. Both start their journey at the beginning of the year from the upper left-hand corner square at Day 0 (being sages, they can comfortably stay in the same square). One decides to follow the Hilbert curve (as in A163357) on his never-ending journey, while the other follows the Peano curve (as in A163336; both walks are illustrated in entry A166043). This sequence gives the days when they will meet, when they both arrive at the same square on the same day.

From the corner, one king walks first towards the east, while the other walks towards the south, so their paths diverge at the beginning. However, about a week later (Day 8), they meet again on square (2,2), two squares south and two squares east of the starting corner. The next day they are both traveling towards the south, so they meet also on Day 9, at square (3,2). After that, they meet briefly three months later (Day 105), and also about three years later (Day 1126), after which they loathe each other so much that they both walk in solitude for the next 18189 (eighteen thousand one hundred and eighty nine) years before they meet again, total of eleven times in just about one month's time (days 6643718-6643752). - Antti Karttunen, Oct 13 2009 [Edited to Hilbert vs Peano by Kevin Ryde, Aug 29 2020]

A. Karttunen, Table of n, a(n) for n = 0..29

Subset of A165480. Cf. also A165465, A163901.

A163897 a(n) = A163531(n)-A163547(n).

Original entry on oeis.org

0, 0, 2, 4, -2, -8, -6, 0, 0, 0, 2, 16, 16, 12, 6, 0, 0, 8, 10, 24, 28, 16, 0, 0, 0, 0, 10, 28, 24, 16, 32, 40, 48, 48, 24, 20, -2, -24, -40, -36, -40, -44, -64, -60, -56, -48, -42, -56, -42, -36, -20, -16, -8, 0, 16, 8, 14, 36, 34, 24, 28, 28, 18, 24, 16, 8, -8, 0, -10

Offset: 0

Antti Karttunen, Sep 19 2009

sign

This sequence gives the difference of squares of distance from the origin to the n-th term, in the Peano curve (A163334) and the Hilbert curve (A163357) on an N x N grid. Because the Hilbert curve is based on powers of 4 and the Peano curve on powers of 9, the graph of this sequence contains dramatic swings. [Edited to Peano vs Hilbert by Kevin Ryde, Aug 28 2020]

Antti Karttunen, Table of n, a(n) for n = 0..65535

Cf. A165480 (positions of 0's).

A163907 Permutation A163905 shown in N x N grid.

Original entry on oeis.org

0, 1, 2, 9, 3, 14, 10, 11, 12, 15, 16, 8, 4, 13, 47, 18, 19, 6, 7, 45, 44, 20, 17, 27, 5, 39, 46, 40, 21, 22, 25, 26, 37, 38, 41, 42, 149, 23, 30, 24, 58, 36, 33, 43, 234, 150, 151, 29, 28, 56, 57, 34, 35, 232, 235, 152, 148, 157, 31, 54, 59, 49, 32, 236, 233, 227, 154

Offset: 0

Antti Karttunen, Sep 19 2009

			The top left 8x8 corner of this array:
+0 +1 +9 10 16 18 20 21
+2 +3 11 +8 19 17 22 23
14 12 +4 +6 27 25 30 29
15 13 +7 +5 26 24 28 31
47 45 39 37 58 56 54 53
44 46 38 36 57 59 52 55
40 41 33 34 49 51 60 61
42 43 35 32 48 50 62 63

Inverse: A163908. a(n) = A163905(A054238(n)) = A163355(A163357(n)). See also A163357, A163917.

A163917 Permutation A163915 shown in N x N grid.

Original entry on oeis.org

0, 1, 3, 7, 2, 9, 5, 6, 8, 10, 16, 4, 14, 11, 48, 17, 18, 13, 12, 51, 50, 20, 19, 28, 15, 52, 49, 60, 21, 23, 29, 31, 53, 55, 61, 63, 127, 22, 27, 30, 47, 54, 57, 62, 149, 125, 126, 25, 24, 46, 45, 59, 56, 148, 150, 118, 124, 113, 26, 39, 44, 35, 58, 152, 151, 146, 117

Offset: 0

Antti Karttunen, Sep 19 2009

			The top left 8x8 corner of this array:
+0 +1 +7 +5 16 17 20 21
+3 +2 +6 +4 18 19 23 22
+9 +8 14 13 28 29 27 25
10 11 12 15 31 30 24 26
48 51 52 53 47 46 39 37
50 49 55 54 45 44 36 38
60 61 57 59 35 34 40 41
63 62 56 58 32 33 43 42

Inverse: A163918. a(n) = A163915(A054238(n)) = A163355(A163907(n)) = A163905(A163357(n)). See also A163357, A163907.

A165466 Squared distance between n's location in A163334 array and A163359 array.

Original entry on oeis.org

0, 2, 2, 2, 2, 10, 10, 2, 0, 0, 2, 10, 20, 10, 10, 18, 32, 32, 50, 74, 100, 100, 72, 50, 32, 50, 50, 34, 20, 20, 16, 16, 16, 10, 4, 4, 2, 4, 8, 8, 8, 10, 20, 18, 20, 20, 26, 50, 50, 40, 20, 20, 20, 20, 32, 32, 34, 40, 58, 74, 100, 74, 74, 80, 80, 80, 52, 52, 50, 34, 34, 32

Offset: 0

Antti Karttunen, Oct 06 2009

nonn

Equivalently, squared distance between n's location in A163336 array and A163357 array. See example at A166043.

A. Karttunen, Table of n, a(n) for n = 0..65535

Positions of zeros: A165467. See also A166043, A165464, A163897, A163900.

a(n) = A000290(abs(A163529(n)-A059253(n))) + A000290(abs(A163528(n)-A059252(n))).

A262174 Sierpiński arrowhead curve as a triangular array starting leftward from the top, read by rows.

Original entry on oeis.org

1, 2, 0, 0, 3, 4, 9, 8, 0, 5, 10, 0, 7, 6, 0, 0, 11, 0, 0, 23, 24, 13, 12, 0, 0, 22, 0, 25, 14, 0, 17, 18, 0, 21, 26, 0, 0, 15, 16, 0, 19, 20, 0, 27, 28, 69, 68, 0, 0, 0, 0, 0, 0, 0, 29, 70, 0, 67, 0, 0, 0, 0, 0, 31, 30, 0, 0, 71, 66, 0, 0, 0, 0, 0, 32, 0, 35, 36

Offset: 1

Max Barrentine, Sep 13 2015

The triangle up to the (1 + 2^n)th row is the n-th iteration of the curve, rotated such that the curve begins at the top and continues down to the left.

As this is not a space-filling curve, not all points on the triangular lattice are reached by the curve; these points are given the value 0.

			The first 5 rows of this triangle show how this curve begins (connect the terms in numerical order):
            1;
          2,  0;
        0,  3,  4;
      9,  8,  0,  5;
   10,  0,  7,  6,  0;
   ...

Max Barrentine, Table of n, a(n) for n = 1..2144
Wikipedia, Sierpiński arrowhead curve

See also A163357, A163334, and A054238 for other fractal curves.

A265318 Fibonacci word fractal in an n X n grid, starting downwards from the top-left corner, listed antidiagonally.

Original entry on oeis.org

1, 0, 2, 5, 3, 0, 6, 4, 0, 0, 7, 0, 0, 0, 0, 0, 8, 10, 0, 0, 20, 0, 0, 9, 11, 0, 19, 21, 0, 0, 0, 0, 12, 18, 0, 22, 0, 0, 0, 0, 13, 0, 17, 23, 0, 0, 0, 0, 0, 0, 14, 16, 0, 24, 26, 0, 0, 0, 0, 0, 0, 15, 0, 0, 25, 27, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 28, 83

Offset: 1

Max Barrentine, Dec 06 2015

The n-th iteration of this curve ends at the n-th Fibonacci number.

As this is not a space-filling curve, not all points on the grid are reached by the curve; these points are given the value 0.

			The top left corner of the array shows how this curve begins (connect the terms in numerical order):
   1   0   5   6   7
   2   3   4   0   8
   0   0   0  10   9
   0   0   0  11   0
   0   0   0  12  13
  20  19  18   0  14
  21   0  17  16  15
  22  23   0   0   0
   0  24   0   0   0
  26  25   0   0   0
  27   0  31  32  33
  28  29  30   0  34

Max Barrentine, Table of n, a(n) for n = 1..2484
Alexis Monnerot-Dumaine, The Fibonacci Word fractal, HAL Id: hal-00367972, 2009.
Alexis Monnerot-Dumaine, The Fibonacci word fractal [Cached copy, with permission]

Cf. A332298, A332299.

See also A163357, A163334, and A054238 for other fractal curves.

cp's OEIS Frontend

A163361 Hilbert curve in N x N grid, one-based, starting rightwards from the top-left corner.

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A165467 Positions of zeros in A165466. Fixed points of A166043/A166044.

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A163897 a(n) = A163531(n)-A163547(n).

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A163907 Permutation A163905 shown in N x N grid.

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A163917 Permutation A163915 shown in N x N grid.

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A165466 Squared distance between n's location in A163334 array and A163359 array.

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A262174 Sierpiński arrowhead curve as a triangular array starting leftward from the top, read by rows.

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A265318 Fibonacci word fractal in an n X n grid, starting downwards from the top-left corner, listed antidiagonally.

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