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

Author

Clark Kimberling, May 31 2013

Keywords

• nonn
• easy

Comments

Zeckendorf distance is defined at A226207.

Examples

			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.
		

Links

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

Crossrefs

Cf. A226080, A226207, A000045.

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