A226215 - OEIS

%I #5 Dec 04 2016 19:46:31

%S 1,2,5,11,18,20,36,45,49,53,69,83,94,105,116,122,122,146,164,178,191,

%T 204,217,229,244,253,265,263,293,309,328,339,357,372,385,400,415,430,

%U 447,462,476,490,504,516,541,536,573,580,600,618,636,654,671,686,704

%N Zeckendorf distance between n! and (n+1)! .

%C Zeckendorf distance is defined at A226207.  For n = 100, the Zeckendorf distance between n! and (n+1)! is 704, compared to the linear distance which exceeds 10^150; in general, the Zeckendorf distance between two large integers tends to be notably less than the ordinary distance.

%H Clark Kimberling, <a href="/A226215/b226215.txt">Table of n, a(n) for n = 1..200</a>

%e 4! = 21 + 3 -> 13 + 2 -> 8 + 1 -> 5 -> 3, and 5! = 89 + 21 + 8 + 2 -> 55 + 13 + 5 + 1 -> 34 + 8 + 3 -> 21 + 5 + 2 -> 13 + 3 + 1 -> 8 + 2 -> 5 + 1 -> 3.   The total number of Zeckendorf downshifts (i.e., arrows) is 4 + 7, so that a(4) = D(24,120) = 11.

%t 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,

%t Min[{Length[z1], Length[z2]}]] - 1)]; lst = Map[d[#!, (#+1)!]] &, Range[100]] (* _Peter J. C. Moses_, May 30 2013 *)

%Y Cf. A226080, A226207.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, May 31 2013