A163531 -id:A163531 - cp's OEIS frontend

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

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).

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), ... .

Original entry on oeis.org

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

A163547 The square of the distance from the origin to the n-th term in the type I Hilbert's Hamiltonian walk A163357.

Original entry on oeis.org

0, 1, 2, 1, 4, 9, 10, 5, 8, 13, 18, 13, 10, 5, 4, 9, 16, 17, 26, 25, 36, 49, 50, 37, 40, 53, 58, 45, 34, 29, 20, 25, 32, 41, 50, 41, 52, 65, 74, 61, 72, 85, 98, 85, 74, 61, 52, 65, 58, 53, 40, 45, 34, 25, 20, 29, 26, 17, 16, 25, 36, 37, 50, 49, 64, 81, 82, 65, 68, 73, 90, 85

Offset: 0

Antti Karttunen, Aug 01 2009

nonn

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

A163530 a(n) = A163528(n)+A163529(n).

Original entry on oeis.org

0, 1, 2, 3, 2, 1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 4, 5, 6, 7, 8, 9, 8, 7, 8, 9, 10, 11, 10, 9, 10, 11, 12, 13, 12, 11, 10, 9, 8, 7, 8, 9, 8, 7, 6, 5, 4, 3, 4, 5, 6, 7, 6, 5, 6, 7, 8, 9, 8, 7, 8, 9, 10, 11, 12, 13, 12, 11, 10, 9, 10, 11, 12, 13, 14, 15, 14, 13, 14, 15, 16, 17, 18, 19, 18, 17

Offset: 0

Antti Karttunen, Aug 01 2009

nonn

A. Karttunen, Table of n, a(n) for n = 0..59048

cp's OEIS Frontend

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

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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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A163547 The square of the distance from the origin to the n-th term in the type I Hilbert's Hamiltonian walk A163357.

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A163530 a(n) = A163528(n)+A163529(n).

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