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A258247 Irregular triangle (Beatty tree for sqrt(8)) as determined in Comments; a permutation of the nonnegative integers. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

LINKS

Table of n, a(n) for n=1..63.

EXAMPLE

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).

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 *)

CROSSREFS

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

Sequence in context: A075647 A334474 A258243 * A085470 A200584 A099379

Adjacent sequences:  A258244 A258245 A258246 * A258248 A258249 A258250

KEYWORD

nonn,tabf,easy

AUTHOR

Clark Kimberling, Jun 08 2015

STATUS

approved

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Last modified July 25 20:18 EDT 2021. Contains 346291 sequences. (Running on oeis4.)