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

0,3

Links

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

Formula

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

Example

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

Mathematica

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 *)

Crossrefs

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.

Sequence in context: A334206 A006666 A267830 * A029683 A063567 A072223

Adjacent sequences: A163331 A163332 A163333 * A163335 A163336 A163337

Keyword

nonn,tabl,look

Author

Antti Karttunen, Jul 29 2009

Extensions

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

Name corrected by Kevin Ryde, Aug 22 2020

Status

approved