A226210 a(n) is the Zeckendorf distance between n and Fibonacci(n).

0, 1, 1, 2, 0, 3, 6, 2, 5, 8, 11, 12, 6, 9, 12, 15, 16, 19, 20, 21, 13, 16, 19, 22, 23, 26, 27, 28, 31, 32, 33, 34, 35, 25, 28, 31, 34, 35, 38, 39, 40, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 45, 48, 51, 54, 55, 58, 59, 60, 63, 64, 65, 66, 67, 70

Offset

1,4

Comments

Zeckendorf distance is defined at A226207.

Links

Clark Kimberling, Table of n, a(n) for n = 1..150

Example

7 = 5 + 2 -> 3 + 1 -> 2, and 13 -> 8 -> 5 -> 3 -> 2.  The total number of Zeckendorf downshifts (i.e., arrows) is 6, so that a(7) = D(7,F(7)) = 6.

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

Crossrefs

Cf. A226080, A226207, A000045.

Sequence in context: A262256 A011120 A256930 * A194737 A071089 A144090

Adjacent sequences: A226207 A226208 A226209 * A226211 A226212 A226213

Keyword

nonn,easy

Author

Clark Kimberling, May 31 2013

Status

approved