A163481 -id:A163481 - 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

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), ...

Original entry on oeis.org

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

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  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

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

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.

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 *)

Name corrected by Kevin Ryde, Aug 28 2020

A163480 Row 0 of A163334 (column 0 of A163336).

Original entry on oeis.org

0, 1, 2, 15, 16, 17, 18, 19, 20, 141, 142, 143, 144, 145, 146, 159, 160, 161, 162, 163, 164, 177, 178, 179, 180, 181, 182, 1275, 1276, 1277, 1278, 1279, 1280, 1293, 1294, 1295, 1296, 1297, 1298, 1311, 1312, 1313, 1314, 1315, 1316, 1437, 1438, 1439

Offset: 0

Antti Karttunen, Jul 29 2009

nonn

Cf. A163481 (Y axis), A037314 (Z-order X axis).

Coordinates: A163528, A163529.

a(n) = my(v=digits(n,3),s=Mod(0,2)); for(i=1,#v, if(s,v[i]+=6); s+=v[i]); fromdigits(v,9); \\ Kevin Ryde, Sep 29 2020

a(n) = A163332(A037314(n)). - Kevin Ryde, Sep 29 2020

A163343 Central diagonal of A163334 and A163336.

Original entry on oeis.org

0, 4, 8, 44, 40, 36, 72, 76, 80, 404, 400, 396, 360, 364, 368, 332, 328, 324, 648, 652, 656, 692, 688, 684, 720, 724, 728, 3644, 3640, 3636, 3600, 3604, 3608, 3572, 3568, 3564, 3240, 3244, 3248, 3284, 3280, 3276, 3312, 3316, 3320, 2996, 2992, 2988

Offset: 0

Antti Karttunen, Jul 29 2009

nonn

It is easy to see by induction that these terms are always divisible by 4.

Cf. A163344 (quarter), A128173, A163332, A163334, A338086.

Peano curve axes: A163480, A163481.

a(n) = my(v=digits(n,3),s=Mod(0,2)); for(i=1,#v, if(s,v[i]=2-v[i]); s+=v[i]); fromdigits(v,9)<<2; \\ Kevin Ryde, Nov 06 2020

a(n) = 4*A163344(n).

a(n) = A163332(A338086(n)) = A338086(A128173(n)). - Kevin Ryde, Nov 06 2020

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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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), ...

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A163480 Row 0 of A163334 (column 0 of A163336).

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A163343 Central diagonal of A163334 and A163336.

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