login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A258243 Irregular triangle (Beatty tree for e) as determined in Comments; a permutation of the nonnegative integers. 2
0, 2, 1, 8, 3, 5, 24, 10, 16, 9, 67, 4, 6, 29, 46, 25, 27, 184, 19, 11, 13, 17, 70, 76, 81, 127, 68, 502, 7, 32, 38, 48, 54, 26, 28, 30, 47, 187, 192, 209, 222, 347, 185, 1367, 12, 14, 18, 20, 21, 84, 89, 106, 133, 149, 69, 71, 73, 77, 78, 82, 128, 130, 505 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The Beatty tree for an irrational number r > 1 (such as r = e), 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 = 20).

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

5   3    24

16  10   9    67

6   46   4    29   27   25   184

19  17   127  13   11   81   76   70   68   502

Generations 0 to 8 of the tree are drawn by the Mathematica program.  In T, the path from 0 to 20 is (0,2,1,5,16,6,19,54,20).  The path obtained by backtracking (i.e., successive applications of the mapping b in Comments) is (20,54,19,6,16,5,1,2,0).

MATHEMATICA

r = E; 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 -> 850]

Map[DeleteDuplicates, Transpose[paths]] (* Peter J. C. Moses, May 21 2015 *)

CROSSREFS

Cf. A022843, A258244 (path-length, 0 to n), A258212

Sequence in context: A082834 A075647 A334474 * A258247 A085470 A200584

Adjacent sequences:  A258240 A258241 A258242 * A258244 A258245 A258246

KEYWORD

nonn,tabf,easy

AUTHOR

Clark Kimberling, Jun 08 2015

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 25 09:15 EDT 2021. Contains 348239 sequences. (Running on oeis4.)