A226211 Zeckendorf distance between n and 2*n.

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

Author

Clark Kimberling, May 31 2013

Keywords

• nonn
• easy

Comments

Zeckendorf distance is defined at A226207.

Examples

			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.
		

Links

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

Crossrefs

Cf. A226080, A226207, A226212.

Programs

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 *)