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A258243
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Irregular triangle (Beatty tree for e) as determined in Comments; a permutation of the nonnegative integers.
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2
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0, 2, 1, 8, 3, 5, 24, 10, 16, 9, 67, 4, 6, 29, 46, 25, 27, 184, 19, 11, 13, 17, 70, 76, 81, 127, 68, 502, 7, 32, 38, 48, 54, 26, 28, 30, 47, 187, 192, 209, 222, 347, 185, 1367, 12, 14, 18, 20, 21, 84, 89, 106, 133, 149, 69, 71, 73, 77, 78, 82, 128, 130, 505
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OFFSET
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1,2
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COMMENTS
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The Beatty tree for an irrational number r > 1 (such as r = e), is formed as follows. To begin, let s = r/(r-1), so that the sequences defined u and v defined by u(n) = floor(r*n) and v(n) = floor(s*n), for n >=1 are the Beatty sequences of r and s, and u and v partition the positive integers.
The tree T has root 0 with an edge to 2, and all other edges are determined as follows: if x is in u(v), then there is an edge from x to floor(r + r*x) and an edge from x to ceiling(x/r); otherwise there is an edge from x to floor(r + r*x). (Thus, the only branchpoints are the numbers in u(v).)
Another way to form T is by "backtracking" to the root 0. Let b(x) = floor[x/r] if x is in (u(n)), and b(x) = floor[r*x] if x is in (v(n)). Starting at any vertex x, repeated applications of b eventually reach 0. The number of steps to reach 0 is the number of the generation of T that contains x. (See Example for x = 20).
See A258212 for a guide to Beatty trees for various choices of r.
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LINKS
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EXAMPLE
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Rows (or generations, or levels) of T:
0
2
1 8
5 3 24
16 10 9 67
6 46 4 29 27 25 184
19 17 127 13 11 81 76 70 68 502
Generations 0 to 8 of the tree are drawn by the Mathematica program. In T, the path from 0 to 20 is (0,2,1,5,16,6,19,54,20). The path obtained by backtracking (i.e., successive applications of the mapping b in Comments) is (20,54,19,6,16,5,1,2,0).
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MATHEMATICA
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r = E; k = 2000; w = Map[Floor[r #] &, Range[k]];
f[x_] := f[x] = If[MemberQ[w, x], Floor[x/r], Floor[r*x]];
b := NestWhileList[f, #, ! # == 0 &] &;
bs = Map[Reverse, Table[b[n], {n, 0, k}]];
generations = Table[DeleteDuplicates[Map[#[[n]] &, Select[bs, Length[#] > n - 1 &]]], {n, 9}]
paths = Sort[Map[Reverse[b[#]] &, Last[generations]]]
graph = DeleteDuplicates[Flatten[Map[Thread[Most[#] -> Rest[#]] &, paths]]]
TreePlot[graph, Top, 0, VertexLabeling -> True, ImageSize -> 850]
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CROSSREFS
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KEYWORD
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nonn,tabf,easy
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AUTHOR
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STATUS
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approved
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