%I
%S 0,1,2,4,3,5,7,4,6,8,10,5,7,9,9,11,6,8,8,10,12,10,12,14,7,9,9,11,11,
%T 13,15,11,13,15,17,8,10,12,10,12,14,12,14,16,18,12,14,16,16,18,9,11,
%U 11,13,15,11,13,15,17,13,15,17,17,19,21,13,15,17,19,17
%N Number of steps from n to 0, where allowable steps are x > [x/r] if x = is in A001951 (the Beatty sequence for sqrt(2)) and x > [r*x] otherwise, where [ ] = floor and r = sqrt(2).
%C a(n) = number of edges from 0 to n in the tree at A258237.
%H Clark Kimberling, <a href="/A258238/b258238.txt">Table of n, a(n) for n = 0..10000</a>
%e 36>25>17>24>16>11>7>4>2>1>0, so that a(36) = 10.
%t r = Sqrt[2]; w = Table[Floor[r*n], {n, 1, 1000}];
%t f[x_] := If[MemberQ[w, x], Floor[x/r], Floor[r*x]];
%t g[x_] := Drop[FixedPointList[f, x], 1];
%t Table[1+ Length[g[n]], {n, 0, 100}]
%Y Cf. A258237, A258212, A001951.
%K nonn,easy
%O 0,3
%A _Clark Kimberling_, Jun 05 2015
