A226214 Zeckendorf distance between n and n^2.

0, 1, 2, 5, 3, 6, 4, 6, 10, 9, 11, 12, 9, 11, 13, 12, 8, 6, 10, 15, 12, 14, 16, 16, 13, 15, 15, 11, 17, 11, 13, 18, 18, 15, 17, 17, 19, 19, 19, 16, 16, 18, 18, 18, 14, 14, 8, 12, 14, 16, 21, 21, 21, 21, 18, 18, 20, 20, 20, 22, 22, 22, 22, 22, 19, 19, 19, 21

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, and 7^2 = 34 + 13 + 2 -> 21 + 8 + 1 -> 13 + 5 -> 8 + 3 -> 5 + 2.  The total number of Zeckendorf downshifts (i.e., arrows) is 4, so that a(7) = D(7,49) = 4.

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.

Sequence in context: A212614 A037852 A367433 * A331212 A160516 A341492

Adjacent sequences: A226211 A226212 A226213 * A226215 A226216 A226217

Keyword

nonn,easy

Author

Clark Kimberling, May 31 2013

Status

approved