A066182 -id:A066182 - cp's OEIS frontend

A194029 Natural fractal sequence of the Fibonacci sequence (1, 2, 3, 5, 8, ...).

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, 29, 30, 31, 32, 33, 34

Offset: 1

Clark Kimberling, Aug 12 2011

Suppose that c(1), c(2), c(3), ... is an 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 defined by:
  If c(k) <= n < c(k+1), then f(n) = 1 + n - c(k).
This defines the present sequence a(n) = f(n) for c = A000045.
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 a different example from the one considered here (c = A000045), 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, ...) = A002260, and a northwest corner of T = A194029 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 / 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
This is true, because the sequence c = A000045 has the property that c(k+1)  - c(k) = c(k-1), so the number of integers {1, 2, 3, ...} to be filled in from index n = c(k) to n = c(k+1)-1 is equal to c(k-1); see also the first EXAMPLE. - M. F. Hasler, Apr 23 2022

			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)

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

Alois P. Heinz, Rows n = 1..20, flattened
Clark Kimberling, Numeration systems and fractal sequences, Acta Arithmetica 73 (1995) 103-117.

Cf. A000045 (Fibonacci numbers).
Cf. A066628, A194030, A194031 (natural interspersion of A000045 and inverse permutation).
Cf. A130853.

T:= n-> $1..(<<0|1>, <1|1>>^n)[1, 2]:
seq(T(n), n=1..10);  # Alois P. Heinz, Dec 11 2024

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 *)

a(n) = A066628(n)+1. - Alan Michael Gómez Calderón, Oct 30 2023

Edited by M. F. Hasler, Apr 23 2022

A066181 Permutation of the integers with cycle form {1}, {2, 3}, {4, 5, 6}, {7, 8, 9, 10}, ...

Original entry on oeis.org

1, 3, 2, 5, 6, 4, 8, 9, 10, 7, 12, 13, 14, 15, 11, 17, 18, 19, 20, 21, 16, 23, 24, 25, 26, 27, 28, 22, 30, 31, 32, 33, 34, 35, 36, 29, 38, 39, 40, 41, 42, 43, 44, 45, 37, 47, 48, 49, 50, 51, 52, 53, 54, 55, 46, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 56, 68, 69, 70, 71, 72, 73

Offset: 1

Wouter Meeussen, Dec 15 2001

Arrange natural numbers 1,2,3,4,5,... as a triangle like A000027, then rotate each row of triangle one left. - Antti Karttunen, May 07 2002

For a group of n terms a^(n)(k)= k where n(n-1)/2 < k <= n(n+1)/2. E.g. for the three terms 5, 6 and 4, a(5)= 6, a(6) = 4, a(4) = 5; a(a(a(5))) = 5 = a^(3)(5). - Amarnath Murthy, May 31 2003

Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A066182.

a := proc(n) floor(sqrt(2*n));n+1;`if`(2*n=%%*(%%+1),%-%%,%) end: # Peter Luschny, Jan 25 2017

FromCycles[Table[n(n-1)/2+Range[n], {n, 13}]]

a(n) = 1 + n + binomial(round(sqrt(2*n)),2) - binomial(round(sqrt(2*n+2)),2). - Brian Tenneson, Jan 23 2017

A097288 Rectangular array read by antidiagonals obtained from A097287 by deleting the zeros.

Original entry on oeis.org

2, 1, 6, 3, 4, 12, 7, 5, 8, 20, 13, 9, 10, 14, 30, 21, 15, 11, 16, 22, 42, 31, 23, 17, 18, 24, 32, 56, 43, 33, 25, 19, 26, 34, 44, 72, 57, 45, 35, 27, 28, 36, 46, 58, 90, 73, 59, 47, 37, 29, 38, 48, 60, 74, 110, 91, 75, 61, 49, 39, 40, 50, 62, 76, 92, 132, 111, 93, 77, 63, 51

Offset: 1

Clark Kimberling, Aug 05 2004

As a sequence, this is a permutation of the natural numbers. Evens lie on or below the main diagonal; odds, above. The triangle formed by the evens is twice that of A066182.

			Northwest corner:
2 1 3 7 13
6 4 5 9 15
12 8 10 11 17
20 14 16 18 19

Cf. A097287.

cp's OEIS Frontend

A194029 Natural fractal sequence of the Fibonacci sequence (1, 2, 3, 5, 8, ...).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A066181 Permutation of the integers with cycle form {1}, {2, 3}, {4, 5, 6}, {7, 8, 9, 10}, ...

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A097288 Rectangular array read by antidiagonals obtained from A097287 by deleting the zeros.

Views

Author

Keywords

Comments

Examples

Crossrefs