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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.
16
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
1,3
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
2+sqrt(2) A258239 A258240
EXAMPLE
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).
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 *)
CROSSREFS
Cf. A000201, A001950, A258212 (path-length from 0 to n).
Sequence in context: A248971 A329435 A160795 * A347704 A355019 A092401
KEYWORD
nonn,tabf,easy
AUTHOR
Clark Kimberling, Jun 05 2015
STATUS
approved