

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
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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)^24*k=Q*n^2 and related dispersions, Journal of Integer Sequences 10 (2007, Article 07.2.7) 117.


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



