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A194029 Natural fractal sequence of the Fibonacci sequence (1,2,3,5,8,...). 37
1, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Suppose that c(1), c(2), c(3),... is a strictly increasing sequence of positive integers with c(1)=1, and that the sequence c(k+1)-c(k) is strictly increasing.  The natural fractal sequence f of c is here introduced by the following rule:

...

If c(k)<=n<c(k+1), then f(n)=1+n-c(k).

...

The natural interspersion of c is here introduced as the array given by T(n,k)=(position of k-th n in f).  Note that c=(row 1 of T).

...

As an example, let c=A000217=(1,3,6,10,15,...), the triangular numbers, so that

  f=(1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,2,3,4,5,6,1,...), and a northwest corner of T is

  ...

  1  3  6 10 15

  2  4  7 11 16

  5  8 12 17 23

  9 13 18 24 31

  ...

Since every number in the set N of positive integers occurs exactly once in this (and every) interspersion, a listing of the terms of T by antidiagonals comprises a permutation, p, of N; letting q denote the inverse of p, we thus have for each c a fractal sequence, an interspersion T, and two permutations of N:

...

c..........f..........T, as p....q......

A000045....A194029....A194030....A194031

A000290....A071797....A194032....A194033

A000217....A002260....A066182....A066181

A028387....A074294....A194034....A194035

A028872....A071797....A194036....A194037

A034856....A002260....A194038....A194040

It appears that this is also a triangle read by rows in which row n lists the first A000045(n) positive integers, n >= 1 (see example). - Omar E. Pol, May 28 2012

REFERENCES

Clark Kimberling, "Fractal sequences and interspersions," Ars Combinatoria 45 (1997) 157-168.

LINKS

Table of n, a(n) for n=1..82.

Clark Kimberling, Numeration systems and fractal sequences, Acta Arithmetica 73 (1995) 103-117.

EXAMPLE

The sequence (1,2,3,5,8,13,...) is used to place 1's in positions numbered 1,2,3,5,8,13,...  Then gaps are filled in with consecutive counting numbers:

  1,1,1,2,1,2,3,1,2,3,4,5,1,...

From Omar E. Pol, May 28 2012: (Start)

Written as an irregular triangle the sequence begins:

1;

1;

1,2;

1,2,3;

1,2,3,4,5;

1,2,3,4,5,6,7,8;

1,2,3,4,5,6,7,8,9,10,11,12,13;

1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21;

The row lengths are A000045(n).

(End)

MATHEMATICA

z = 40;

c[k_] := Fibonacci[k + 1];

c = Table[c[k], {k, 1, z}]  (* A000045 *)

f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]

f = Table[f[n], {n, 1, 800}]  (* A194029 *)

r[n_] := Flatten[Position[f, n]]

t[n_, k_] := r[n][[k]]

TableForm[Table[t[n, k], {n, 1, 8}, {k, 1, 7}]]

p = Flatten[Table[t[k, n - k + 1], {n, 1, 13}, {k, 1, n}]]  (* A194030 *)

q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 80}]]  (* A194031 *)

Flatten[Range[Fibonacci[Range[66]]]] (* Birkas Gyorgy, Jun 30 2012 *)

CROSSREFS

Cf. A000045, A194030, A194031.

Sequence in context: A194844 A138528 A037125 * A194055 A162192 A238954

Adjacent sequences:  A194026 A194027 A194028 * A194030 A194031 A194032

KEYWORD

nonn,tabf

AUTHOR

Clark Kimberling, Aug 12 2011

STATUS

approved

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Last modified June 23 21:29 EDT 2017. Contains 288675 sequences.