A226211 Zeckendorf distance between n and 2n.

1, 1, 1, 4, 3, 5, 6, 5, 7, 7, 8, 8, 7, 7, 9, 9, 8, 10, 10, 10, 9, 9, 9, 11, 11, 11, 11, 10, 12, 12, 12, 12, 12, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 13, 12, 12, 14, 14, 14, 14, 14, 14, 14, 14, 13, 13, 13, 13, 13, 13, 13, 15, 15, 15, 15, 15, 15, 15, 15, 15

Offset

1,4

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 22 = 21 + 1 -> 13 -> 8 -> 5 -> 3 -> 2.  The total number of Zeckendorf downshifts (i.e., arrows) is 8, so that a(11) = D(11,22) = 8.

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[#, 2# &, Range[100]] (* Peter J. C. Moses, May 30 2013 *)

Crossrefs

Cf. A226080, A226207, A226212.

Sequence in context: A170816 A336067 A133981 * A296413 A016701 A023829

Adjacent sequences: A226208 A226209 A226210 * A226212 A226213 A226214

Keyword

nonn,easy

Author

Clark Kimberling, May 31 2013

Status

approved