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A258241 Irregular triangle (Beatty tree for r = sqrt(3)), T, of all nonnegative integers, each exactly once, as determined in Comments. 2
0, 1, 3, 2, 6, 4, 5, 12, 7, 8, 10, 22, 15, 19, 13, 39, 9, 11, 24, 27, 23, 34, 69, 14, 16, 17, 20, 48, 60, 40, 41, 43, 121, 31, 36, 25, 28, 29, 35, 72, 76, 84, 105, 70, 71, 211, 18, 21, 45, 51, 64, 42, 44, 49, 50, 55, 61, 62, 126, 147, 183, 122, 124, 133, 367 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Suppose that r is an irrational number > 1, and let s = r/(r-1), so that the sequences 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 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(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 = 8).

See A258212 for a guide to Beatty trees for various choices of r.

LINKS

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

EXAMPLE

Rows (or generations, or levels) of T:

0

1

3

2   6

4   5   12

7   8   10  22

15  19  13  39

9   11  24  27  23  34  69

14  16  17  20  48  60  40  41  43  121

Generations 0 to 10 of the tree are drawn by the Mathematica program.  In T, the path from 0 to 16 is (0,1,3,6,4,8,15,27,16).  The path obtained by backtracking (i.e., successive applications of the mapping b in Comments) is (16,27,15,8,4,6,3,1,0).

MATHEMATICA

r = Sqrt[3]; 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, 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]] (* Peter J. C. Moses, May 21 2015 *)

CROSSREFS

Cf. A022838, A258242 (path-length, 0 to n), A258212

Sequence in context: A191428 A191733 A191444 * A256739 A267104 A093050

Adjacent sequences:  A258238 A258239 A258240 * A258242 A258243 A258244

KEYWORD

nonn,tabf,easy

AUTHOR

Clark Kimberling, Jun 05 2015

STATUS

approved

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Last modified November 28 13:47 EST 2021. Contains 349413 sequences. (Running on oeis4.)