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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), ... .

%I A163334 #39 Feb 16 2025 08:33:11
%S A163334 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,
%T A163334 52,54,19,23,11,43,39,51,55,59,20,22,24,42,40,38,56,58,60,141,21,25,
%U A163334 29,41,37,69,57,61,425,142,140,26,28,30,36,70,68,62,424,426,143,139
%N 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), ... .
%H A163334 Antti Karttunen, <a href="/A163334/b163334.txt">Table of n, a(n) for n = 0..13202</a>
%H A163334 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 (and in figure 3 rotate -90 degrees for the table here).
%H A163334 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 A163334 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HilbertCurve.html">Hilbert curve</a> (this curve called "Hilbert II").
%H A163334 Wikipedia, <a href="http://en.wikipedia.org/wiki/Self-avoiding_walk">Self-avoiding walk</a>
%H A163334 Wikipedia, <a href="http://en.wikipedia.org/wiki/Space-filling_curve">Space-filling curve</a>
%H A163334 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A163334 a(n) = A163332(A163328(n)).
%e A163334 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 A163334    0  1  2 15 16 17 18 19 20
%e A163334    5  4  3 14 13 12 23 22 21
%e A163334    6  7  8  9 10 11 24 25 26
%e A163334   47 46 45 44 43 42 29 28 27
%e A163334   48 49 50 39 40 41 30 31 32
%e A163334   53 52 51 38 37 36 35 34 33
%e A163334   54 55 56 69 70 71 72 73 74
%e A163334   59 58 57 68 67 66 77 76 75
%e A163334   60 61 62 63 64 65 78 79 80
%t A163334 b[{n_, k_}, {m_}] := (A[k, n] = m - 1);
%t A163334 MapIndexed[b, List @@ PeanoCurve[4][[1]]];
%t A163334 Table[A[n - k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Mar 07 2021 *)
%Y A163334 Transpose: A163336. Inverse: A163335. One-based version: A163338. Row sums: A163342. Row 0: A163480. Column 0: A163481. Central diagonal: A163343.
%Y A163334 See also: A163528-A163529, A163531, A163534, A163536, A163897.
%Y A163334 See A163357 and A163359 for the Hilbert curve.
%K A163334 nonn,tabl,look
%O A163334 0,3
%A A163334 _Antti Karttunen_, Jul 29 2009
%E A163334 Links to further derived sequences added by _Antti Karttunen_, Sep 21 2009
%E A163334 Name corrected by _Kevin Ryde_, Aug 22 2020