A163336 Peano curve in an n X n grid, starting downwards 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, 5, 1, 6, 4, 2, 47, 7, 3, 15, 48, 46, 8, 14, 16, 53, 49, 45, 9, 13, 17, 54, 52, 50, 44, 10, 12, 18, 59, 55, 51, 39, 43, 11, 23, 19, 60, 58, 56, 38, 40, 42, 24, 22, 20, 425, 61, 57, 69, 37, 41, 29, 25, 21, 141, 426, 424, 62, 68, 70, 36, 30, 28, 26, 140, 142, 431, 427

Offset

0,2

Links

A. Karttunen, Table of n, a(n) for n = 0..3320

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 (figure 3 with Y downwards is 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).

Rémy Sigrist, Colored scatterplot of (x, y) such that 0 <= x, y < 3^6 (where the hue is function of T(x, y))

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

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  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

Mathematica

b[{n_, k_}, {m_}] := (A[n, k] = 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: A163334. Inverse: A163337. a(n) = A163332(A163330(n)) = A163327(A163333(A163328(n))) = A163334(A061579(n)). One-based version: A163340. Row sums: A163342. Row 0: A163481. Column 0: A163480. Central diagonal: A163343.

See A163357 and A163359 for the Hilbert curve.

Sequence in context: A379131 A086231 A201419 * A173898 A343344 A200644

Adjacent sequences: A163333 A163334 A163335 * A163337 A163338 A163339

Keyword

nonn,tabl

Author

Antti Karttunen, Jul 29 2009

Extensions

Name corrected by Kevin Ryde, Aug 28 2020

Status

approved