A166041 -id:A166041 - cp's OEIS frontend

A165465 Positions of zeros in A165464. Fixed points of A166041/A166042.

0, 1, 7, 8, 15, 16, 22, 23, 24, 25, 1702855, 1702856, 1702857, 1702872, 1702873, 2220150, 3327583, 3329174, 3329270, 3329271, 3329279

Offset: 0

Antti Karttunen, Oct 06 2009

Consider two immortal sage kings traveling on the infinite chessboard, visiting every square at the 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 the day zero (being sages, they can comfortably stay in the same square without bloodshed). One decides to follow the Hilbert curve on his never-ending journey, while the other follows the Peano curve. (These are both illustrated in the entry A166041.) This sequence gives the days when they will meet, when they both come to the same square on the same day.
Both walk first one square towards east, where they meet at Day 1. Then one turns south, while the other one proceeds to the east. However, just six days later, on Day 7, they meet again, at the square (2,1), two squares south and one square east of the starting corner. They also meet the next day (Day 8), as well as another week later (Day 15), and before January is over, they meet still five more times, on Days 16, 22, 23, 24 and 25. However, it takes 4662 years and about three months before they meet again, on three successive days (Days 1702855, 1702856 and 1702857). - Antti Karttunen, Oct 13 2009 [Edited to Hilbert vs Peano by Kevin Ryde, Aug 30 2020]
Subset of A165480. - Antti Karttunen, Oct 13 2009

Antti Karttunen, Table of n, a(n) for n = 0..20

Cf. A165467, A165480, A163901.

A166043 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 A163336.

Original entry on oeis.org

0, 3, 4, 7, 2, 1, 14, 13, 8, 9, 54, 55, 56, 57, 6, 5, 58, 59, 60, 63, 64, 67, 62, 61, 50, 49, 68, 69, 48, 51, 46, 47, 122, 123, 44, 45, 34, 35, 28, 31, 32, 33, 52, 53, 10, 11, 12, 15, 16, 17, 30, 29, 18, 19, 20, 23, 24, 25, 22, 21, 234, 233, 230, 229, 218, 217, 38, 37, 26

Offset: 0

Antti Karttunen, Oct 06 2009

nonn

Fixed points are quite rare: A165467.

			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
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
3 is in position (2,1) in A163336, while A163357(2,1) = 7. Thus a(3) = 7.

Inverse: A166044. a(n) = A163357(A163337(n)) = A163359(A163335(n)). Cf. also A166041.

A166042 Permutation of nonnegative integers: a(n) tells which integer is in the same position in the square array A163334 as where n is located in the array A163357.

Original entry on oeis.org

0, 1, 4, 5, 6, 47, 46, 7, 8, 45, 44, 9, 14, 3, 2, 15, 16, 13, 12, 17, 18, 19, 22, 23, 24, 25, 28, 29, 42, 11, 10, 43, 40, 37, 36, 41, 30, 31, 34, 35, 72, 73, 76, 77, 66, 71, 70, 67, 68, 57, 56, 69, 38, 39, 50, 51, 52, 49, 48, 53, 54, 55, 58, 59, 60, 425, 424, 61, 62, 63, 422

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 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
9 is in position (3,2) in A163357, while A163334(3,2) = 45. Thus a(9) = 45.

Inverse: A166041. a(n) = A163334(A163358(n)) = A163336(A163360(n)). Fixed points: A165465. Cf. also A166044.

A165464 Squared distance between n's location in A163334 array and A163357 array.

Original entry on oeis.org

0, 0, 2, 4, 2, 4, 2, 0, 0, 2, 2, 4, 4, 4, 2, 0, 0, 2, 2, 4, 4, 2, 0, 0, 0, 0, 2, 4, 4, 2, 8, 10, 16, 16, 4, 2, 2, 10, 16, 10, 8, 8, 20, 20, 20, 18, 18, 32, 18, 10, 4, 2, 4, 10, 8, 2, 2, 10, 10, 4, 4, 4, 2, 10, 16, 26, 20, 10, 2, 4, 10, 18, 32, 32, 50, 52, 52, 34, 40, 58, 80, 80, 106, 146

Offset: 0

Antti Karttunen, Oct 06 2009

nonn

Equivalently, squared distance between n's location in A163336 array and A163359 array. See example at A166041.

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

Positions of zeros: A165465. See also A165466, A163897, A163900.

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

cp's OEIS Frontend

A165465 Positions of zeros in A165464. Fixed points of A166041/A166042.

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A166043 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 A163336.

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A166042 Permutation of nonnegative integers: a(n) tells which integer is in the same position in the square array A163334 as where n is located in the array A163357.

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A165464 Squared distance between n's location in A163334 array and A163357 array.

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