A166043 -id:A166043 - cp's OEIS frontend

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

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.

A166041 Permutation of nonnegative integers: a(n) tells which integer is in the same position in the square array A163357 as where n is located in the array A163334.

Original entry on oeis.org

0, 1, 14, 13, 2, 3, 4, 7, 8, 11, 30, 29, 18, 17, 12, 15, 16, 19, 20, 21, 234, 233, 22, 23, 24, 25, 230, 229, 26, 27, 36, 37, 218, 217, 38, 39, 34, 33, 52, 53, 32, 35, 28, 31, 10, 9, 6, 5, 58, 57, 54, 55, 56, 59, 60, 61, 50, 49, 62, 63, 64, 67, 68, 69, 122, 123, 44, 47, 48

Offset: 0

Antti Karttunen, Oct 06 2009

nonn

Fixed points are quite rare: A165465.

			The top left 9 X 9 corner of A163334:
   0  1  2 15 16 17 18 19 20
   5  4  3 14 13 12 23 22 21
   6  7  8  9 10 11 24 25 26
  47 46 45 44 43 42 29 28 27
  48 49 50 39 40 41 30 31 32
  53 52 51 38 37 36 35 34 33
  54 55 56 69 70 71 72 73 74
  59 58 57 68 67 66 77 76 75
  60 61 62 63 64 65 78 79 80
The top left 8 X 8 corner of A163357:
   0  1 14 15 16 19 20 21
   3  2 13 12 17 18 23 22
   4  7  8 11 30 29 24 25
   5  6  9 10 31 28 27 26
  58 57 54 53 32 35 36 37
  59 56 55 52 33 34 39 38
  60 61 50 51 46 45 40 41
  63 62 49 48 47 44 43 42
9 is in position (2,3) in A163334, while A163357(2,3) = 11. Thus a(9) = 11.

Inverse: A166042. a(n) = A163357(A163335(n)) = A163359(A163337(n)). Cf. also A166043.

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))).

A166044 Permutation of nonnegative integers: a(n) tells which integer is in the same position in the square array A163336 as where n is located in the array A163357.

Original entry on oeis.org

0, 5, 4, 1, 2, 15, 14, 3, 8, 9, 44, 45, 46, 7, 6, 47, 48, 49, 52, 53, 54, 59, 58, 55, 56, 57, 68, 69, 38, 51, 50, 39, 40, 41, 36, 37, 70, 67, 66, 71, 72, 77, 76, 73, 34, 35, 30, 31, 28, 25, 24, 29, 42, 43, 10, 11, 12, 13, 16, 17, 18, 23, 22, 19, 20, 141, 140, 21, 26, 27, 134

Offset: 0

Antti Karttunen, Oct 06 2009

nonn

			The top left 8 X 8 corner of A163357:
   0  1 14 15 16 19 20 21
   3  2 13 12 17 18 23 22
   4  7  8 11 30 29 24 25
   5  6  9 10 31 28 27 26
  58 57 54 53 32 35 36 37
  59 56 55 52 33 34 39 38
  60 61 50 51 46 45 40 41
  63 62 49 48 47 44 43 42
The top left 9 X 9 corner of A163336:
   0  5  6 47 48 53 54 59 60
   1  4  7 46 49 52 55 58 61
   2  3  8 45 50 51 56 57 62
  15 14  9 44 39 38 69 68 63
  16 13 10 43 40 37 70 67 64
  17 12 11 42 41 36 71 66 65
  18 23 24 29 30 35 72 77 78
  19 22 25 28 31 34 73 76 79
  20 21 26 27 32 33 74 75 80
12 is in position (1,3) in A163357, while A163336(1,3) = 46. Thus a(12) = 46.

Inverse: A166043. a(n) = A163336(A163358(n)) = A163334(A163360(n)). Fixed points: A165467. Cf. also A166042.

cp's OEIS Frontend

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

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A166041 Permutation of nonnegative integers: a(n) tells which integer is in the same position in the square array A163357 as where n is located in the array A163334.

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

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A166044 Permutation of nonnegative integers: a(n) tells which integer is in the same position in the square array A163336 as where n is located in the array A163357.

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