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A258237 Irregular triangle (Beatty tree for r = sqrt(2)), T, of all nonnegative integers, each exactly once, as described in Comments. 2
0, 1, 2, 4, 3, 7, 5, 11, 8, 16, 6, 12, 24, 9, 18, 17, 35, 14, 13, 25, 26, 50, 10, 19, 21, 38, 36, 72, 15, 28, 27, 31, 52, 51, 55, 103, 20, 22, 37, 39, 41, 45, 73, 74, 79, 147, 32, 29, 56, 53, 59, 65, 106, 104, 113, 209, 23, 42, 40, 46, 76, 75, 80, 84, 93 (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 = 17).

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

2

4

3   7

5   1

8   16

6   12  24

9   18  17  35

14  13  25  26  50

10  19  21  38  36  72

15  28  27  31  52  51  55  103

Generations 0 to 10 of the tree are drawn by the Mathematica program.  In T, the path from 0 to 36 is (0,1,2,4,7,11,16,24,17,25,36).  The path obtained by backtracking (i.e., successive applications of the mapping b in Comments) is (36,25,17,24,16,11,7,4,2,1,0).

MATHEMATICA

r = Sqrt[2]; 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]] (* Peter J. C. Moses, May 21 2015 *)

CROSSREFS

Cf. A001951, A001952, A258238 (path-length from 0 to n), A258212

Sequence in context: A252460 A252752 A056536 * A108228 A127008 A199535

Adjacent sequences:  A258234 A258235 A258236 * A258238 A258239 A258240

KEYWORD

nonn,tabf,easy

AUTHOR

Clark Kimberling, Jun 05 2015

STATUS

approved

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Last modified December 1 20:41 EST 2021. Contains 349435 sequences. (Running on oeis4.)