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A258212
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Irregular triangle (or "lower Wythoff tree", or Beatty tree for r = golden ratio ), T, of all nonnegative integers, each exactly once, as determined from the lower Wythoff sequence as described in Comments.
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16
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0, 1, 3, 2, 6, 4, 11, 8, 7, 19, 5, 12, 14, 32, 9, 21, 24, 20, 53, 16, 13, 15, 33, 35, 40, 87, 10, 22, 25, 27, 55, 58, 66, 54, 142, 17, 37, 42, 45, 34, 36, 41, 88, 90, 95, 108, 231, 29, 23, 26, 28, 56, 59, 61, 67, 69, 74, 144, 147, 155, 176, 143, 375, 18, 38
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OFFSET
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1,3
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COMMENTS
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Let r = (1+ sqrt(5))/2 = golden ratio. Let u(n) = floor[n*r] and v(n) = floor[n*r^2], so that u = (u(n)) = A000201 = lower Wythoff sequence and v = (v(n)) = A001950 = upper Wythoff sequence; it is well known that 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, and b(x) = floor[r*x] if x is in v. 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 = 35).
In the procedure just described, r can be any irrational number > 1. Beatty trees and backtracking sequences for selected r are indicated here:
r Beatty tree for r backtracking sequence, (b(n))
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LINKS
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EXAMPLE
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Rows (or generations, or levels) of T:
0
1
3
6 2
11 4
19 7 8
32 12 14 5
53 20 21 24 9
87 33 35 13 40 15 16
Generations 0 to 10 of the tree are drawn by the Mathematica program. In T, the path from 0 to 35 is (0,1,3,6,11,7,12,21,35). The path obtained by backtracking (i.e., successive applications of the mapping b in Comments) is (35,21,12,7,11,6,3,1,0).
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MATHEMATICA
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r = GoldenRatio; k = 1000; 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]
Map[DeleteDuplicates, Transpose[paths]] (*The numbers in each level of the tree*)
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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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