A258245 Irregular triangle (Beatty tree for Pi) as determined in Comments; a permutation of the nonnegative integers.

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

Offset: 1

Author

Clark Kimberling, Jun 08 2015

Keywords

• nonn
• tabf
• easy

Comments

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.

Examples

			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).
		

Crossrefs

Cf. A022844, A258244 (path-length, 0 to n), A258212.

Programs

Mathematica

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]
Map[DeleteDuplicates, Transpose[paths]] (* Peter J. C. Moses,May 21 2015 *)