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approved

editing

proposed

allocated Irregular triangle (Beatty tree for Clark Kimberlingr = sqrt(2)), T, of all nonnegative integers, each exactly once, as described in Comments.

0, 1, 2, 4, 3, 7, 5, 11, 8, 16, 6, 12, 24, 9, 18, 17, 35, 14, 13, 25, 26, 50, 10, 19, 21, 38, 36, 72, 15, 28, 27, 31, 52, 51, 55, 103, 20, 22, 37, 39, 41, 45, 73, 74, 79, 147, 32, 29, 56, 53, 59, 65, 106, 104, 113, 209, 23, 42, 40, 46, 76, 75, 80, 84, 93

1,3

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 = 17).

See A258212 for a guide to Beatty trees for various choices of r.

Rows (or generations, or levels) of T:

0

1

2

4

3 7

5 1

8 16

6 12 24

9 18 17 35

14 13 25 26 50

10 19 21 38 36 72

15 28 27 31 52 51 55 103

Generations 0 to 10 of the tree are drawn by the Mathematica program. In T, the path from 0 to 36 is (0,1,2,4,7,11,16,24,17,25,36). The path obtained by backtracking (i.e., successive applications of the mapping b in Comments) is (36,25,17,24,16,11,7,4,2,1,0).

r = Sqrt[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, 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]] (* Peter J. C. Moses, May 21 2015 *)

Cf. A001951, A001952, A258238 (path-length from 0 to n), A258212

allocated

nonn,tabf,easy

Clark Kimberling, Jun 05 2015

approved

editing

allocated for Clark Kimberling

allocated

approved