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%I #21 Aug 18 2018 08:33:13
%S 1,2,0,1,1,4,8,9,1,1,6,8,2,4,8,3,6,0,4,9,5,6,6,1,7,4,1,9,0,1,1,1,7,1,
%T 4,7,6,1,6,6,7,1,0,9,0,2,3,5,5,2,7,4,2,3,1,6,1,3,5,1,2,3,0,9,5,4,5,1,
%U 0,4,1,6,7,5,6,4,6,6,3,5,7,6,9,0,0,7,6,8,5,8,3,9,2,8,0,3,1,9,8,0,0,3,0,4,1
%N The spiral of Champernowne, read along the East ray.
%C Inspired by Stanislaw Ulam's spiral, circa 1963.
%H Robert G. Wilson v, <a href="/A244677/a244677.jpg">Cover of the March 1964 issue of Scientific American</a>
%F Formulas for rays in directions of 32 compass points:
%F SE 4n^2 -4n +1
%F SExS 64n^2 -113n +50
%F SSE 16n^2 -25n +10
%F SxE 64n^2 -115n +52
%F S 4n^2 -5n +2
%F SxW 64n^2 -117n +54
%F SSW 16n^2 -27n +12
%F SWxS 64n^2 -119n +56
%F SW 4n^2 -6n +3
%F SWxW 64n^2 -121n +58
%F WSW 16n^2 -29n +14
%F WxS 64n^2 -123n +60
%F W 4n^2 -7n +4
%F WxN 64n^2 -125n +62
%F WNW 16n^2 -31n +16
%F NWxW 64n^2 -127n +64
%F NW 4n^2 -8n +5
%F NWxN 64n^2 -129n +66
%F NNW 16n^2 -33n +18
%F NxW 64n^2 -131n +68
%F N 4n^2 -9n +6
%F NxE 64n^2 -133n +70
%F NNE 16n^2 -35n +20
%F NExN 64n^2 -135n +72
%F NE 4n^2 -10n +7
%F NExE 64n^2 -137n +74
%F ENE 16n^2 -37n +22
%F ExN 64n^2 -139n +76
%F E 4n^2 -11n +8
%F ExS 64n^2 -141n +78
%F ESE 16n^2 -39n +24
%F SExE 64n^2 -143n +80
%e The beginning of the infinite spiral of David Gawen Champernowne:
%e .
%e 7--1--9--6--1--8--6--1--7--6--1--6--6--1--5--6--1--4--6--1--3 .
%e | | |
%e 0 1--4--4--1--3--4--1--2--4--1--1--4--1--0--4--1--9--3--1 6 .
%e | | | | |
%e 1 4 2--1--1--2--1--0--2--1--9--1--1--8--1--1--7--1--1 8 1 .
%e | | | | | | |
%e 7 5 2 0--1--1--0--1--0--0--1--9--9--8--9--7--9--6 6 3 2 9
%e | | | | | | | | |
%e 1 1 1 2 7--7--6--7--5--7--4--7--3--7--2--7--1 9 1 1 6 8
%e | | | | | | | | | | |
%e 1 4 2 1 7 5--5--4--5--3--5--2--5--1--5--0 7 5 1 7 1 1
%e | | | | | | | | | | | | |
%e 7 6 3 0 8 5 7--3--6--3--5--3--4--3--3 5 0 9 5 3 1 8
%e | | | | | | | | | | | | | | |
%e 2 1 1 3 7 6 3 3--2--2--2--1--2--0 3 9 7 4 1 1 6 8
%e | | | | | | | | | | | | | | | | |
%e 1 4 2 1 9 5 8 2 3--1--2--1--1 2 2 4 9 9 1 6 1 1
%e | | | | | | | | | | | | | | | | | | |
%e 7 7 4 0 8 7 3 4 1 5--4--3 1 9 3 8 6 3 4 3 0 7
%e | | | | | | | | | | | | | | | | | | | | |
%e 3 1 1 4 0 5 9 2 4 6 1--2 0 1 1 4 8 9 1 1 6 8
%e | | | | | | | | | | | | | | | | | | | |
%e 1 4 2 1 8 8 4 5 1 7--8--9--1 8 3 7 6 2 1 5 1 1
%e | | | | | | | | | | | | | | | | | |
%e 7 8 5 0 1 5 0 2 5--1--6--1--7--1 0 4 7 9 3 3 9 6
%e | | | | | | | | | | | | | | | |
%e 4 1 1 5 8 9 4 6--2--7--2--8--2--9--3 6 6 1 1 1 5 8
%e | | | | | | | | | | | | | |
%e 1 4 2 1 2 6 1--4--2--4--3--4--4--4--5--4 6 9 1 4 1 1
%e | | | | | | | | | | | |
%e 7 9 6 0 8 0--6--1--6--2--6--3--6--4--6--5--6 0 2 3 8 5
%e | | | | | | | | | |
%e 5 1 1 6 3--8--4--8--5--8--6--8--7--8--8--8--9--9 1 1 5 8
%e | | | | | | | |
%e 1 5 2 1--0--7--1--0--8--1--0--9--1--1--0--1--1--1--1 3 1 1
%e | | | | | |
%e 7 0 7--1--2--8--1--2--9--1--3--0--1--3--1--1--3--2--1--3 7 4
%e | | | |
%e 6 1--5--1--1--5--2--1--5--3--1--5--4--1--5--5--1--5--6--1--5 8
%e | |
%e 1--7--7--1--7--8--1--7--9--1--8--0--1--8--1--1--8--2--1--8--3--1
%t almostNatural[n_, b_] := Block[{m = 0, d = n, i = 1, l, p}, While[m <= d, l = m; m = (b - 1) i*b^(i - 1) + l; i++]; i--; p = Mod[d - l, i]; q = Floor[(d - l)/i] + b^(i - 1); If[p != 0, IntegerDigits[q, b][[p]], Mod[q - 1, b]]]; f[n_] := 4n^2 - 11n + 8 (* see formula section *); Array[ almostNatural[ f@#, 10] &, 105]
%Y Cf. A033307, A054552, A244678 - A244688, A033952, A244690 - A244692.
%K nonn,easy
%O 1,2
%A _Robert G. Wilson v_, Jul 04 2014