A258245 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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.

LINKS

Table of n, a(n) for n=1..62.

EXAMPLE

Rows (or generations, or levels) of T:

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).

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 *)

CROSSREFS

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