A258247 Irregular triangle (Beatty tree for sqrt(8)) as determined in Comments; a permutation of the nonnegative integers.

0, 2, 1, 8, 3, 5, 25, 11, 16, 9, 73, 4, 6, 28, 33, 48, 26, 209, 19, 10, 12, 14, 17, 76, 82, 96, 138, 74, 593, 7, 36, 42, 50, 56, 27, 29, 31, 34, 49, 212, 217, 234, 274, 393, 210, 1680, 13, 15, 18, 20, 22, 84, 90, 98, 104, 121, 144, 161, 75, 77, 79, 83, 97

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 = sqrt(8)), 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 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 = 13).

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

Examples

			Rows (or generations, or levels) of T:
0
2
1   8
3   5   25
11  16  9   73
4   6   28  33  48  26  209
19  10  12  14  16  76  82  96  138  74  593
Generations 0 to 8 of the tree are drawn by the Mathematica program.  In T, the path from 0 to 13 is (0,2,8,3,11,33,12,36,13).  The path obtained by backtracking (i.e., successive applications of the mapping b in Comments) is (13,36,12,33,11,3,8,2,0).
		

Crossrefs

Cf. A022842, A258248 (path-length, 0 to n), A258212.

Programs

Mathematica

r = Sqrt[8]; 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, 9}]
paths = Sort[Map[Reverse[b[#]] &, Last[generations]]]
graph = DeleteDuplicates[Flatten[Map[Thread[Most[#] -> Rest[#]] &, paths]]]
TreePlot[graph, Top, 0, VertexLabeling -> True, ImageSize -> 900]
Map[DeleteDuplicates, Transpose[paths]] (* Peter J. C. Moses,May 21 2015 *)