A226212 Zeckendorf distance between n and floor(n/2).

1, 1, 2, 1, 2, 1, 3, 4, 4, 3, 5, 5, 4, 6, 6, 5, 5, 7, 7, 7, 6, 8, 8, 8, 8, 7, 7, 7, 9, 9, 9, 9, 9, 8, 8, 10, 10, 10, 10, 10, 10, 9, 9, 9, 9, 9, 11, 11, 11, 11, 11, 11, 11, 11, 10, 10, 10, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 11, 11, 11, 11, 11, 11, 11, 11

Offset

1,3

Comments

Zeckendorf distance is defined at A226207.

Links

Clark Kimberling, Table of n, a(n) for n = 1..1000

Example

11 = 8 + 3 -> 5 + 2 -> 3 + 1 -> 2, and 5 -> 3 -> 2.  The total number of Zeckendorf downshifts (i.e., arrows) is 5, so that a(11) = D(11,5) = 5.

Mathematica

zeck[n_Integer] := Block[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]], t = n, z = {}},    While[k > 1, If[t >= Fibonacci[k], AppendTo[z, 1]; t = t - Fibonacci[k], AppendTo[z, 0]]; k--]; If[n > 0 && z[[1]] == 0, Rest[z], z]]; d[n1_, n2_] := Module[{z1 = zeck[n1], z2 = zeck[n2]}, Length[z1] + Length[z2] - 2 (NestWhile[# + 1 &, 1, z1[[#]] == z2[[#]] &, 1,
Min[{Length[z1], Length[z2]}]] - 1)]; lst = Map[d[#, Floor[#/2]] &, Range[100]] (* Peter J. C. Moses, May 30 2013 *)

Crossrefs

Cf. A226080, A226207, A226211.

Sequence in context: A344594 A210850 A051276 * A233439 A256600 A137752

Adjacent sequences: A226209 A226210 A226211 * A226213 A226214 A226215

Keyword

nonn,easy

Author

Clark Kimberling, May 31 2013

Status

approved