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A258239 Irregular triangle (Beatty tree for r = 2 + sqrt(2)), T, of all nonnegative integers, each exactly once, as determined in Comments.

%I A258239 #4 Jun 07 2015 18:03:18
%S A258239 0,3,1,13,6,4,47,2,17,23,14,163,5,7,10,51,61,81,48,559,27,37,15,18,20,
%T A258239 24,167,177,211,279,164,1911,8,11,54,64,71,85,95,129,49,52,62,82,563,
%U A258239 573,607,723,955,560,6527,30,40,16,19,21,25,28,38,170,180,187
%N A258239 Irregular triangle (Beatty tree for r = 2 + sqrt(2)), T, of all nonnegative integers, each exactly once, as determined in Comments.
%C A258239 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.
%C A258239 The tree T has root 0 with an edge to 3, 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).)
%C A258239 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).
%C A258239 See A258212 for a guide to Beatty trees for various choices of r.
%e A258239 Rows (or generations, or levels) of T:
%e A258239 0
%e A258239 3
%e A258239 1   13
%e A258239 6   4   47
%e A258239 2   23  17  14   163
%e A258239 10  7   81  5    61   51   48   559
%e A258239 37  27  24  279  20   18   211  15   177   167  164   1911
%e A258239 Generations 0 to 7 of the tree are drawn by the Mathematica program.  In T, the path from 0 to 8 is (0,3,1,6,23,7,27,8).  The path obtained by backtracking (i.e., successive applications of the mapping b in Comments) is (8,27,7,23,6,1,3,0).
%t A258239 r = 2+Sqrt[2]; k = 1000; w = Map[Floor[r #] &, Range[k]];
%t A258239 f[x_] := f[x] = If[MemberQ[w, x], Floor[x/r], Floor[r*x]];
%t A258239 b := NestWhileList[f, #, ! # == 0 &] &;
%t A258239 bs = Map[Reverse, Table[b[n], {n, 0, k}]];
%t A258239 generations = Table[DeleteDuplicates[Map[#[[n]] &, Select[bs, Length[#] > n - 1 &]]], {n, 8}]
%t A258239 paths = Sort[Map[Reverse[b[#]] &, Last[generations]]]
%t A258239 graph = DeleteDuplicates[Flatten[Map[Thread[Most[#] -> Rest[#]] &, paths]]]
%t A258239 TreePlot[graph, Top, 0, VertexLabeling -> True, ImageSize -> 700]
%t A258239 Map[DeleteDuplicates, Transpose[paths]] (* _Peter J. C. Moses_,May 21 2015 *)
%Y A258239 Cf. A001951, A001952, A258240 (path-length, 0 to n), A258212
%K A258239 nonn,tabf,easy
%O A258239 1,2
%A A258239 _Clark Kimberling_, Jun 05 2015