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A120874 Fractal sequence of the Fraenkel array (A038150).

%I A120874 #18 Feb 13 2015 09:08:42
%S A120874 1,2,1,3,4,2,5,1,6,7,3,8,9,4,10,2,11,12,5,13,1,14,15,6,16,17,7,18,3,
%T A120874 19,20,8,21,22,9,23,4,24,25,10,26,2,27,28,11,29,30,12,31,5,32,33,13,
%U A120874 34,1,35,36,14,37,38,15,39,6,40,41,16,42,43,17,44,7,45,46,18,47,3,48,49,19
%N A120874 Fractal sequence of the Fraenkel array (A038150).
%C A120874 A fractal sequence f contains itself as a proper subsequence; e.g., if you delete the first occurrence of each positive integer, the remaining sequence is f; thus f properly contains itself infinitely many times.
%D A120874 Clark Kimberling, The equation (j+k+1)^2-4*k=Q*n^2 and related dispersions, Journal of Integer Sequences 10 (2007, Article 07.2.7) 1-17.
%H A120874 N. J. A. Sloane, <a href="/classic.html#WYTH">Classic Sequences</a>.
%e A120874 The fractal sequence f(n) of a dispersion D={d(g,h,)} is defined as follows. For each positive integer n there is a unique (g,h) such that n=d(g,h) and f(n)=g. So f(6)=2 because the row of the Fraenkel array in which 6 occurs is row 2.
%t A120874 num[n_, b_] := Last[NestWhile[{Mod[#[[1]], Last[#[[2]]]], Drop[#[[2]], -1], Append[#[[3]], Quotient[#[[1]], Last[#[[2]]]]]} &, {n, b, {}}, #[[2]] =!= {} &]];
%t A120874 left[n_, b_] := If[Last[num[n, b]] == 0, Dot[num[n, b], Rest[Append[Reverse[b], 0]]], n];
%t A120874 fractal[n_, b_] := # - Count[Last[num[Range[#], b]], 0] &@ FixedPoint[left[#, b] &, n];
%t A120874 Table[fractal[n, Table[Fibonacci[2 i], {i, 12}]], {n, 30}] (* _Birkas Gyorgy_, Apr 13 2011 *)
%t A120874 Table[Ceiling[NestWhile[Ceiling[#/GoldenRatio^2] - 1 &, n, Ceiling[#/GoldenRatio] == Ceiling[(# - 1)/GoldenRatio]&]/ GoldenRatio], {n, 30}] (* _Birkas Gyorgy_, Apr 15 2011 *)
%Y A120874 Cf. A038150.
%K A120874 nonn
%O A120874 1,2
%A A120874 _Clark Kimberling_, Jul 10 2006