A226209 Zeckendorf distance between n and n+2.

2, 1, 1, 3, 4, 3, 5, 4, 6, 6, 7, 7, 4, 6, 6, 8, 8, 4, 9, 9, 4, 6, 6, 8, 8, 4, 10, 10, 4, 6, 6, 11, 11, 4, 6, 6, 8, 8, 4, 10, 10, 4, 6, 6, 12, 12, 4, 6, 6, 8, 8, 4, 13, 13, 4, 6, 6, 8, 8, 4, 10, 10, 4, 6, 6, 12, 12, 4, 6, 6, 8, 8, 4, 14, 14, 4, 6, 6, 8, 8, 4

Offset

1,1

Comments

Zeckendorf distance is defined at A226207.

Links

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

Example

7 = 5 + 2 -> 3 + 1 -> 2, and 9 = 8 + 1 -> 5 -> 3 -> 2.  The total number of Zeckendorf downshifts (i.e., arrows) is 5, so that a(7) = D(7,9) = 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[#, # + 2] &, Range[100]] (* Peter J. C. Moses, May 30 2013 *)

Crossrefs

Cf. A226080, A226207, A226208.

Sequence in context: A371279 A100529 A262953 * A302097 A307277 A210691

Adjacent sequences: A226206 A226207 A226208 * A226210 A226211 A226212

Keyword

nonn,easy

Author

Clark Kimberling, May 31 2013

Status

approved