A226208 Zeckendorf distance between n and n+1.

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

Offset

1,3

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

Crossrefs

Cf. A226080.

Sequence in context: A121998 A120499 A027749 * A304743 A214540 A214595

Adjacent sequences: A226205 A226206 A226207 * A226209 A226210 A226211

Keyword

nonn,easy

Author

Clark Kimberling, May 31 2013

Status

approved