A163359 - OEIS

%I #21 Mar 07 2021 17:53:12

%S 0,3,1,4,2,14,5,7,13,15,58,6,8,12,16,59,57,9,11,17,19,60,56,54,10,30,

%T 18,20,63,61,55,53,31,29,23,21,64,62,50,52,32,28,24,22,234,65,67,49,

%U 51,33,35,27,25,233,235,78,66,68,48,46,34,36,26,230,232,236,79,77,71

%N Hilbert curve in N x N grid, starting downwards from the top-left corner, listed by descending antidiagonals.

%H A. Karttunen, <a href="/A163359/b163359.txt">Table of n, a(n) for n = 0..32895</a>

%H David Hilbert, <a href="https://doi.org/10.1007/BF01199431">Ueber die stetige Abbildung einer Linie auf ein Flächenstück</a>, Mathematische Annalen, volume 38, number 3, 1891, pages 459-460.  Also <a href="https://eudml.org/doc/157555">EUDML</a> (link to GDZ).

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

%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 nonnegative integers</a>

%e The top left 8x8 corner of the array shows how this surjective self-avoiding walk begins (connect the terms in numerical order, 0-1-2-3-...):

%e    +0 +3 +4 +5 58 59 60 63

%e    +1 +2 +7 +6 57 56 61 62

%e    14 13 +8 +9 54 55 50 49

%e    15 12 11 10 53 52 51 48

%e    16 17 30 31 32 33 46 47

%e    19 18 29 28 35 34 45 44

%e    20 23 24 27 36 39 40 43

%e    21 22 25 26 37 38 41 42

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

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

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

%Y Transpose: A163357, a(n) = A163357(A061579(n)). Inverse: A163360. One-based version: A163363. Row sums: A163365. Row 0: A163483. Column 0: A163482. Central diagonal: A062880.

%Y See also A163334 and A163336 for the Peano curve.

%K nonn,tabl

%O 0,2

%A _Antti Karttunen_, Jul 29 2009