

A258235


Irregular triangle (or "upper Wythoff tree", or Beatty tree for r = (golden ratio)^2), T, of all nonnegative integers, each exactly once, as determined from the upper Wythoff sequence as described in Comments.


2



0, 2, 1, 7, 3, 5, 20, 10, 15, 8, 54, 4, 6, 23, 28, 41, 21, 143, 9, 11, 13, 16, 18, 57, 62, 75, 109, 55, 376, 31, 36, 44, 49, 22, 24, 26, 29, 42, 146, 151, 164, 198, 287, 144, 986, 12, 14, 17, 19, 65, 70, 78, 83, 96, 112, 117, 130, 56, 58, 60, 63, 76, 110
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OFFSET

1,2


COMMENTS

Suppose that r is an irrational number > 1, and let s = r/(r1), 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 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 = 29).
See A258212 for a guide to Beatty trees for various choices of r.


LINKS

Table of n, a(n) for n=1..63.


EXAMPLE

Rows (or generations, or levels) of T:
0
2
1 7
3 5 20
10 15 8 54
4 6 23 28 41 21 143
9 11 13 16 18 57 62 75 109 55 376
Generations 0 to 9 of the tree are drawn by the Mathematica program. In T, the path from 0 to 29 is (0,2,7,3,10,28,75,29). The path obtained by backtracking (i.e., successive applications of the mapping b in Comments) is (29,75,28,10,3,7,2,0).


MATHEMATICA

r = (3 + Sqrt[5])/2; 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, 10}]
paths = Sort[Map[Reverse[b[#]] &, Last[generations]]]
graph = DeleteDuplicates[Flatten[Map[Thread[Most[#] > Rest[#]] &, paths]]]
TreePlot[graph, Top, 0, VertexLabeling > True, ImageSize > 1400]
Map[DeleteDuplicates, Transpose[paths]] (* Peter J. C. Moses, May 21 2015 *)


CROSSREFS

Cf. A000201, A001950, A258236 (pathlength from 0 to n).
Sequence in context: A180335 A257699 A160535 * A330877 A021050 A194797
Adjacent sequences: A258232 A258233 A258234 * A258236 A258237 A258238


KEYWORD

nonn,tabf,easy


AUTHOR

Clark Kimberling, Jun 05 2015


STATUS

approved



