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A258241
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Irregular triangle (Beatty tree for r = sqrt(3)), T, of all nonnegative integers, each exactly once, as determined in Comments.
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2
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0, 1, 3, 2, 6, 4, 5, 12, 7, 8, 10, 22, 15, 19, 13, 39, 9, 11, 24, 27, 23, 34, 69, 14, 16, 17, 20, 48, 60, 40, 41, 43, 121, 31, 36, 25, 28, 29, 35, 72, 76, 84, 105, 70, 71, 211, 18, 21, 45, 51, 64, 42, 44, 49, 50, 55, 61, 62, 126, 147, 183, 122, 124, 133, 367
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OFFSET
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1,3
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COMMENTS
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Suppose that r is an irrational number > 1, and let s = r/(r-1), so that the sequences 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 1, 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
1
3
2 6
4 5 12
7 8 10 22
15 19 13 39
9 11 24 27 23 34 69
14 16 17 20 48 60 40 41 43 121
Generations 0 to 10 of the tree are drawn by the Mathematica program. In T, the path from 0 to 16 is (0,1,3,6,4,8,15,27,16). The path obtained by backtracking (i.e., successive applications of the mapping b in Comments) is (16,27,15,8,4,6,3,1,0).
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MATHEMATICA
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r = Sqrt[3]; 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, 11}]
paths = Sort[Map[Reverse[b[#]] &, Last[generations]]]
graph = DeleteDuplicates[Flatten[Map[Thread[Most[#] -> Rest[#]] &, paths]]]
TreePlot[graph, Top, 0, VertexLabeling -> True, ImageSize -> 700]
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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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