A163335 -id:A163335 - cp's OEIS frontend

A163334 Peano curve in an n X n grid, starting rightwards from the top left corner, listed antidiagonally as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ... .

0, 1, 5, 2, 4, 6, 15, 3, 7, 47, 16, 14, 8, 46, 48, 17, 13, 9, 45, 49, 53, 18, 12, 10, 44, 50, 52, 54, 19, 23, 11, 43, 39, 51, 55, 59, 20, 22, 24, 42, 40, 38, 56, 58, 60, 141, 21, 25, 29, 41, 37, 69, 57, 61, 425, 142, 140, 26, 28, 30, 36, 70, 68, 62, 424, 426, 143, 139

Offset: 0

Antti Karttunen, Jul 29 2009

			The top left 9 X 9 corner of the array shows how this surjective self-avoiding walk begins (connect the terms in numerical order, 0-1-2-3-...):
   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

Antti Karttunen, Table of n, a(n) for n = 0..13202
E. H. Moore, On Certain Crinkly Curves, Transactions of the American Mathematical Society, volume 1, number 1, 1900, pages 72-90. (And errata.) See section 7 (and in figure 3 rotate -90 degrees for the table here).
Giuseppe Peano, Sur une courbe, qui remplit toute une aire plane, Mathematische Annalen, volume 36, number 1, 1890, pages 157-160. Also EUDML (link to GDZ).
Eric Weisstein's World of Mathematics, Hilbert curve (this curve called "Hilbert II").
Wikipedia, Self-avoiding walk
Wikipedia, Space-filling curve
Index entries for sequences that are permutations of the natural numbers

Transpose: A163336. Inverse: A163335. One-based version: A163338. Row sums: A163342. Row 0: A163480. Column 0: A163481. Central diagonal: A163343.
See also: A163528-A163529, A163531, A163534, A163536, A163897.
See A163357 and A163359 for the Hilbert curve.

b[{n_, k_}, {m_}] := (A[k, n] = m - 1);
MapIndexed[b, List @@ PeanoCurve[4][[1]]];
Table[A[n - k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Mar 07 2021 *)

a(n) = A163332(A163328(n)).

Links to further derived sequences added by Antti Karttunen, Sep 21 2009

Name corrected by Kevin Ryde, Aug 22 2020

A163528 The X-coordinate of the n-th point in the Peano curve A163334.

Original entry on oeis.org

0, 1, 2, 2, 1, 0, 0, 1, 2, 3, 4, 5, 5, 4, 3, 3, 4, 5, 6, 7, 8, 8, 7, 6, 6, 7, 8, 8, 7, 6, 6, 7, 8, 8, 7, 6, 5, 4, 3, 3, 4, 5, 5, 4, 3, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 3, 4, 5, 5, 4, 3, 3, 4, 5, 6, 7, 8, 8, 7, 6, 6, 7, 8, 9, 10, 11, 11, 10, 9, 9, 10, 11, 12, 13, 14, 14, 13, 12

Offset: 0

Antti Karttunen, Aug 01 2009

nonn

There is a 2-state automaton that accepts exactly those pairs (n,a(n)) where n is represented in base 9 and a(n) in base 3; see accompanying file a163528.pdf - Jeffrey Shallit, Aug 10 2023

Antti Karttunen, Table of n, a(n) for n = 0..59048
Jeffrey Shallit, Automaton for A163528
Index entries for sequences related to coordinates of 2D curves

Cf. A025581, A163335, A002262, A163337, A163325, A163332.

See also: A059252, A059253, A163529, A163530, A163531, A163532.

a(n) = A025581(A163335(n)) = A002262(A163337(n)) = A163325(A163332(n)).

Name corrected by Kevin Ryde, Aug 28 2020

A163529 The Y-coordinate of the n-th point in the Peano curve A163334.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 4, 4, 4, 3, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 8, 8, 7, 7, 7, 6, 6, 6, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 8, 8, 7, 7, 7, 6, 6, 6, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 8, 8

Offset: 0

Antti Karttunen, Aug 01 2009

There is a 2-state automaton that accepts exactly those pairs (n,a(n)) where n is represented in base 9 and a(n) in base 3; see accompanying file a163529.pdf. - Jeffrey Shallit, Aug 10 2023

Antti Karttunen, Table of n, a(n) for n = 0..59048
Jeffrey Shallit, Automaton for A163529
Index entries for sequences related to coordinates of 2D curves

Cf. A002262, A163335, A025581, A163337, A163326, A163332.

Cf. A059252, A059253, A163528, A163530, A163531, A163533.

a(n) = A002262(A163335(n)) = A025581(A163337(n)) = A163326(A163332(n)).

Name corrected by Kevin Ryde, Aug 28 2020

A163337 Inverse permutation to A163336.

Original entry on oeis.org

0, 2, 5, 8, 4, 1, 3, 7, 12, 18, 25, 33, 26, 19, 13, 9, 14, 20, 27, 35, 44, 53, 43, 34, 42, 52, 63, 74, 62, 51, 61, 73, 86, 99, 85, 72, 60, 49, 39, 31, 40, 50, 41, 32, 24, 17, 11, 6, 10, 16, 23, 30, 22, 15, 21, 29, 38, 47, 37, 28, 36, 46, 57, 69, 82, 96, 83, 70, 58, 48, 59, 71

Offset: 0

Antti Karttunen, Jul 29 2009

nonn

abs(A025581(a(n+1))-A025581(a(n))) + abs(A002262(a(n+1))-A002262(a(n))) = 1 for all n.

Inverse: A163336. a(n) = A163331(A163332(n)) = A061579(A163335(n)). One-based version: A163341. See also A163335, A163358.

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.

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.

A163339 Inverse permutation to A163338.

Original entry on oeis.org

1, 2, 4, 8, 5, 3, 6, 9, 13, 18, 24, 31, 23, 17, 12, 7, 11, 16, 22, 29, 37, 47, 38, 30, 39, 48, 58, 70, 59, 49, 60, 71, 83, 97, 84, 72, 61, 51, 42, 33, 41, 50, 40, 32, 25, 19, 14, 10, 15, 20, 26, 34, 27, 21, 28, 35, 43, 53, 44, 36, 45, 54, 64, 75, 87, 100, 86, 74, 63, 52, 62

Offset: 1

Antti Karttunen, Jul 29 2009

nonn

Inverse: A163338.

a(n) = A163335(n-1) + 1.

cp's OEIS Frontend

A163334 Peano curve in an n X n grid, starting rightwards from the top left corner, listed antidiagonally as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ... .

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A163528 The X-coordinate of the n-th point in the Peano curve A163334.

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A163529 The Y-coordinate of the n-th point in the Peano curve A163334.

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A163337 Inverse permutation to A163336.

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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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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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A163339 Inverse permutation to A163338.

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