A163336 - OEIS

%I #20 Mar 08 2021 06:16:04

%S 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,

%T 12,18,59,55,51,39,43,11,23,19,60,58,56,38,40,42,24,22,20,425,61,57,

%U 69,37,41,29,25,21,141,426,424,62,68,70,36,30,28,26,140,142,431,427

%N 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), ...

%H A. Karttunen, <a href="/A163336/b163336.txt">Table of n, a(n) for n = 0..3320</a>

%H E. H. Moore, <a href="https://doi.org/10.1090/S0002-9947-1900-1500526-4">On Certain Crinkly Curves</a>, Transactions of the American Mathematical Society, volume 1, number 1, 1900, pages 72-90.  (And <a href="https://doi.org/10.1090/S0002-9947-1900-1500428-3/">errata</a>.)  See section 7 (figure 3 with Y downwards is the table here).

%H Giuseppe Peano, <a href="https://doi.org/10.1007/BF01199438">Sur une courbe, qui remplit toute une aire plane</a>, Mathematische Annalen, volume 36, number 1, 1890, pages 157-160.  Also <a href="https://eudml.org/doc/157489">EUDML</a> (link to GDZ).

%H Rémy Sigrist, <a href="/A163336/a163336.png">Colored scatterplot of (x, y) such that 0 <= x, y < 3^6</a> (where the hue is function of T(x, y))

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HilbertCurve.html">Hilbert curve</a> (this curve called "Hilbert II").

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Self-avoiding_walk">Self-avoiding walk</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Space-filling_curve">Space-filling curve</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%e 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-...):

%e    0  5  6 47 48 53 54 59 60

%e    1  4  7 46 49 52 55 58 61

%e    2  3  8 45 50 51 56 57 62

%e   15 14  9 44 39 38 69 68 63

%e   16 13 10 43 40 37 70 67 64

%e   17 12 11 42 41 36 71 66 65

%e   18 23 24 29 30 35 72 77 78

%e   19 22 25 28 31 34 73 76 79

%e   20 21 26 27 32 33 74 75 80

%t b[{n_, k_}, {m_}] := (A[n, k] = m - 1);

%t MapIndexed[b, List @@ PeanoCurve[4][[1]]];

%t Table[A[n - k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Mar 07 2021 *)

%Y 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.

%Y See A163357 and A163359 for the Hilbert curve.

%K nonn,tabl

%O 0,2

%A _Antti Karttunen_, Jul 29 2009

%E Name corrected by _Kevin Ryde_, Aug 28 2020