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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), ...

%I A163336 #21 Feb 16 2025 08:33:11
%S A163336 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 A163336 12,18,59,55,51,39,43,11,23,19,60,58,56,38,40,42,24,22,20,425,61,57,
%U A163336 69,37,41,29,25,21,141,426,424,62,68,70,36,30,28,26,140,142,431,427
%N 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), ...
%H A163336 A. Karttunen, <a href="/A163336/b163336.txt">Table of n, a(n) for n = 0..3320</a>
%H A163336 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 A163336 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 A163336 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 A163336 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HilbertCurve.html">Hilbert curve</a> (this curve called "Hilbert II").
%H A163336 Wikipedia, <a href="http://en.wikipedia.org/wiki/Self-avoiding_walk">Self-avoiding walk</a>
%H A163336 Wikipedia, <a href="http://en.wikipedia.org/wiki/Space-filling_curve">Space-filling curve</a>
%H A163336 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e A163336 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 A163336    0  5  6 47 48 53 54 59 60
%e A163336    1  4  7 46 49 52 55 58 61
%e A163336    2  3  8 45 50 51 56 57 62
%e A163336   15 14  9 44 39 38 69 68 63
%e A163336   16 13 10 43 40 37 70 67 64
%e A163336   17 12 11 42 41 36 71 66 65
%e A163336   18 23 24 29 30 35 72 77 78
%e A163336   19 22 25 28 31 34 73 76 79
%e A163336   20 21 26 27 32 33 74 75 80
%t A163336 b[{n_, k_}, {m_}] := (A[n, k] = m - 1);
%t A163336 MapIndexed[b, List @@ PeanoCurve[4][[1]]];
%t A163336 Table[A[n - k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Mar 07 2021 *)
%Y A163336 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 A163336 See A163357 and A163359 for the Hilbert curve.
%K A163336 nonn,tabl
%O A163336 0,2
%A A163336 _Antti Karttunen_, Jul 29 2009
%E A163336 Name corrected by _Kevin Ryde_, Aug 28 2020