A120874 Fractal sequence of the Fraenkel array (A038150).

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, 19, 20, 8, 21, 22, 9, 23, 4, 24, 25, 10, 26, 2, 27, 28, 11, 29, 30, 12, 31, 5, 32, 33, 13, 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

Offset: 1

Author

Clark Kimberling, Jul 10 2006

Keywords

• nonn

Comments

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.

Examples

			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.
		

References

• 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.

Links

• N. J. A. Sloane, Classic Sequences.

Crossrefs

Cf. A038150.

Programs

Mathematica

num[n_, b_] := Last[NestWhile[{Mod[#[[1]], Last[#[[2]]]], Drop[#[[2]], -1], Append[#[[3]], Quotient[#[[1]], Last[#[[2]]]]]} &, {n, b, {}}, #[[2]] =!= {} &]];
left[n_, b_] := If[Last[num[n, b]] == 0, Dot[num[n, b], Rest[Append[Reverse[b], 0]]], n];
fractal[n_, b_] := # - Count[Last[num[Range[#], b]], 0] &@ FixedPoint[left[#, b] &, n];
Table[fractal[n, Table[Fibonacci[2 i], {i, 12}]], {n, 30}] (* Birkas Gyorgy, Apr 13 2011 *)
Table[Ceiling[NestWhile[Ceiling[#/GoldenRatio^2] - 1 &, n, Ceiling[#/GoldenRatio] == Ceiling[(# - 1)/GoldenRatio]&]/ GoldenRatio], {n, 30}] (* Birkas Gyorgy, Apr 15 2011 *)