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A258245
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Irregular triangle (Beatty tree for Pi) as determined in Comments; a permutation of the nonnegative integers.
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2
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0, 3, 1, 12, 6, 4, 40, 2, 15, 21, 13, 128, 5, 7, 9, 43, 50, 69, 41, 405, 25, 31, 14, 16, 18, 22, 131, 138, 160, 219, 129, 1275, 8, 10, 53, 59, 72, 81, 100, 42, 44, 47, 51, 70, 408, 414, 436, 505, 691, 406, 4008, 34, 17, 19, 23, 26, 28, 32, 141, 150, 163, 169
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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 = Pi), 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 3, 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 = 8).
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
3
1 12
6 4 40
2 21 15 13 128
9 7 69 5 50 43 42 405
31 25 22 219 18 16 160 14 138 131 129 1275
Generations 0 to 7 of the tree are drawn by the Mathematica program. In T, the path from 0 to 8 is (0,3,1,6,21,7,25,8). The path obtained by backtracking (i.e., successive applications of the mapping b in Comments) is (8,25,7,21,6,1,3,0).
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MATHEMATICA
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r = Pi; 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, 8}]
paths = Sort[Map[Reverse[b[#]] &, Last[generations]]]
graph = DeleteDuplicates[Flatten[Map[Thread[Most[#] -> Rest[#]] &, paths]]]
TreePlot[graph, Top, 0, VertexLabeling -> True, ImageSize -> 800]
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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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