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

sqrt(2)          A258237              A258238

2+sqrt(2)        A258239              A258240

sqrt(3)          A258241              A258242

e                A258243              A258244

Pi               A258245              A258246

sqrt(8)          A258247              A258248

LINKS

Table of n, a(n) for n=1..65.

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: A091018 A248971 A160795 * A092401 A222208 A116626

Adjacent sequences:  A258209 A258210 A258211 * A258213 A258214 A258215

KEYWORD

nonn,tabf,easy

AUTHOR

Clark Kimberling, Jun 05 2015

STATUS

approved

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Last modified August 17 06:13 EDT 2017. Contains 290635 sequences.