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A120874 Fractal sequence of the Fraenkel array (A038150). 0
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 (list; graph; refs; listen; history; text; internal format)
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: A208750 A107893 A131987 * A112382 A117384 A125160

Adjacent sequences:  A120871 A120872 A120873 * A120875 A120876 A120877

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jul 10 2006

STATUS

approved

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Last modified March 19 17:21 EDT 2019. Contains 321330 sequences. (Running on oeis4.)