A258212 Irregular triangle (or "lower Wythoff tree", or Beatty tree for r = golden ratio ), T, of all nonnegative integers, each exactly once, as determined from the lower Wythoff sequence as described in Comments.

0, 1, 3, 2, 6, 4, 11, 8, 7, 19, 5, 12, 14, 32, 9, 21, 24, 20, 53, 16, 13, 15, 33, 35, 40, 87, 10, 22, 25, 27, 55, 58, 66, 54, 142, 17, 37, 42, 45, 34, 36, 41, 88, 90, 95, 108, 231, 29, 23, 26, 28, 56, 59, 61, 67, 69, 74, 144, 147, 155, 176, 143, 375, 18, 38

Offset: 1

Author

Clark Kimberling, Jun 05 2015

Keywords

• nonn
• tabf
• easy

Comments

Let r = (1+ sqrt(5))/2 = golden ratio. Let u(n) = floor[n*r] and v(n) = floor[n*r^2], so that u = (u(n)) = A000201 = lower Wythoff sequence and v = (v(n)) = A001950 = upper Wythoff sequence; it is well known that 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, and b(x) = floor[r*x] if x is in v. 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 = 35).

In the procedure just described, r can be any irrational number > 1. Beatty trees and backtracking sequences for selected r are indicated here:

r Beatty tree for r backtracking sequence, (b(n))

(1+sqrt(5))/2 A258212 A258215

(3+sqrt(5))/2 A258235 A258236

sqrt(2) A258237 A258238

2+sqrt(2) A258239 A258240

sqrt(3) A258241 A258242

e A258243 A258244

Pi A258245 A258246

sqrt(8) A258247 A258248

Examples

			Rows (or generations, or levels) of T:
0
1
3
6   2
11  4
19  7   8
32  12  14  5
53  20  21  24  9
87  33  35  13  40  15  16
Generations 0 to 10 of the tree are drawn by the Mathematica program.  In T, the path from 0 to 35 is (0,1,3,6,11,7,12,21,35).  The path obtained by backtracking (i.e., successive applications of the mapping b in Comments) is (35,21,12,7,11,6,3,1,0).
		

Crossrefs

Cf. A000201, A001950, A258212 (path-length from 0 to n).

Programs

Mathematica

r = GoldenRatio; 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]] (*The numbers in each level of the tree*)
(* Peter J. C. Moses, May 21 2015 *)