A194833 -id:A194833 - cp's OEIS frontend

A194834 Inverse permutation of A194833; every positive integer occurs exactly once.

1, 2, 3, 6, 4, 5, 9, 7, 10, 8, 13, 11, 14, 12, 15, 18, 21, 16, 19, 17, 20, 24, 27, 22, 25, 28, 23, 26, 36, 31, 34, 29, 32, 35, 30, 33, 44, 39, 42, 37, 45, 40, 43, 38, 41, 53, 48, 51, 46, 54, 49, 52, 47, 55, 50, 63, 58, 66, 61, 56, 64, 59, 62, 57, 65, 60, 74, 69, 77

Offset: 1

Clark Kimberling, Sep 03 2011

nonn

Cf. A194832, A194833.

Mathematica
```
(See A194832.)
```

A194832 Triangular array (and fractal sequence): row n is the permutation of (1,2,...,n) obtained from the increasing ordering of fractional parts {r}, {2r}, ..., {nr}, where r= -tau = -(1+sqrt(5))/2.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 03 2011

Every irrational number r generates a triangular array in the manner exemplified here.  Taken as a sequence, the numbers comprise a fractal sequence f which induces a second (rectangular) array whose n-th row gives the positions of n in f.  Denote these by Array1 and Array2.  As proved elsewhere, Array2 is an interspersion.  (Every row intersperses every other row except for initial terms.)  Taken as a sequence, Array2 is a permutation, Perm1, of the positive integers; let Perm2 denote its inverse permutation.
Examples:
r................Array1....Array2....Perm2
tau..............A054065...A054069...A054068
-tau.............A194832...A194833...A194834
sqrt(2)..........A054073...A054077...A054076
-sqrt(2).........A194835...A194836...A194837
sqrt(3)..........A194838...A194839...A194840
-sqrt(3).........A194841...A194842...A194843
sqrt(5)..........A194844...A194845...A194846
-sqrt(5).........A194856...A194857...A194858
sqrt(6)..........A194871...A194872...A194873
-sqrt(6).........A194874...A194875...A194876
sqrt(8)..........A194877...A194878...A194879
-sqrt(8).........A194896...A194897...A194898
sqrt(12).........A194899...A194900...A194901
-sqrt(12)........A194902...A194903...A194904
e................A194859...A194860...A194861
-e...............A194865...A194866...A194864
pi...............A194905...A194906...A194907
-pi..............A194908...A194909...A194910
(1+sqrt(3))/2....A194862...A194863...A194867
-(1+sqrt(3))/2...A194868...A194869...A194870
2^(1/3)..........A194911...A194912...A194913

			Fractional parts: {-r}=-0.61..;{-2r}=-0.23..;{-3r}=-0.85..;{-4r}=-0.47..; thus, row 4 is (3,1,4,2) because {-3r} < {-r} < {-4r} < {-2r}. [corrected by _Michel Dekking_, Nov 30 2020]
First nine rows:
  1
  1 2
  3 1 2
  3 1 4 2
  3 1 4 2 5
  3 6 1 4 2 5
  3 6 1 4 7 2 5
  8 3 6 1 4 7 2 5
  8 3 6 1 9 4 7 2 5

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

Wikipedia, Fractal sequence

Cf. A194833, A194834, A054065.

r = -GoldenRatio;
t[n_] := Table[FractionalPart[k*r], {k, 1, n}];
f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@ Sort[t[n], Less]], {n, 1, 20}]]
(* A194832 *)
TableForm[Table[Flatten[(Position[t[n], #1] &) /@ Sort[t[n], Less]], {n, 1, 15}]]
row[n_] := Position[f, n];
u = TableForm[Table[row[n], {n, 1, 20}]]
g[n_, k_] := Part[row[n], k];
p = Flatten[Table[g[k, n - k + 1], {n, 1, 13}, {k, 1, n}]] (* A194833 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 80}]] (* A194834 *)

Table in overview corrected by Georg Fischer, Jul 30 2023

A372231 Fixed points of A372341.

Original entry on oeis.org

1, 2, 5, 8, 12, 18, 24, 32, 40, 49, 60, 71, 83, 97, 111, 127, 143, 160, 179, 198, 219, 240, 262, 286, 310, 335, 362, 389, 418, 447, 477, 509, 541, 574, 609, 644, 681, 718, 756, 796, 836, 878, 920, 963, 1008, 1053, 1099, 1147, 1195, 1245, 1295, 1346, 1399, 1452

Offset: 1

Rémy Sigrist, Apr 28 2024

nonn

			A372341(49) = 49, so 49 belongs to this sequence.

Rémy Sigrist, C++ program

Cf. A000217, A005206, A194832, A194833, A372341.

a(n) = A000217(n) - A005206(n-1).

A194832(a(n)) = 1. Also the first row A194833(1, n) = a(n). This can be seen in the EXAMPLE section of A372341. Each row is horizontally shifted by a value from A005206. The shift is known to be floor((k+1)*tau)-k-1, where tau is the golden ratio. The position of "1" in a permutation from A194832 is determined by a similar process based on the same constant. - Thomas Scheuerle, Jul 08 2024

cp's OEIS Frontend

A194834 Inverse permutation of A194833; every positive integer occurs exactly once.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A194832 Triangular array (and fractal sequence): row n is the permutation of (1,2,...,n) obtained from the increasing ordering of fractional parts {r}, {2r}, ..., {nr}, where r= -tau = -(1+sqrt(5))/2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

A372231 Fixed points of A372341.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula