A258238 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).

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, 13, 15, 11, 13, 15, 17, 8, 10, 12, 10, 12, 14, 12, 14, 16, 18, 12, 14, 16, 16, 18, 9, 11, 11, 13, 15, 11, 13, 15, 17, 13, 15, 17, 17, 19, 21, 13, 15, 17, 19, 17

Offset: 0

Author

Clark Kimberling, Jun 05 2015

Keywords

• nonn
• easy

Comments

a(n) = number of edges from 0 to n in the tree at A258237.

Examples

			36->25->17->24->16->11->7->4->2->1->0, so that a(36) = 10.
		

Links

• Clark Kimberling, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A258237, A258212, A001951.

Programs

Mathematica

r = Sqrt[2]; w = Table[Floor[r*n], {n, 1, 1000}];
f[x_] := If[MemberQ[w, x], Floor[x/r], Floor[r*x]];
g[x_] := Drop[FixedPointList[f, x], -1];
Table[-1+ Length[g[n]], {n, 0, 100}]