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

1,2

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.

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

Table of n, a(n) for n=1..79.

N. J. A. Sloane, Classic Sequences.

Example

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.

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 *)

Crossrefs

Cf. A038150.

Sequence in context: A107893 A131987 A337226 * A358103 A112382 A117384

Adjacent sequences: A120871 A120872 A120873 * A120875 A120876 A120877

Keyword

nonn

Author

Clark Kimberling, Jul 10 2006

Status

approved