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.

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

Author

Clark Kimberling, Sep 03 2011

Keywords

• nonn
• tabl

Comments

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

Examples

			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
		

References

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

Links

• Wikipedia, Fractal sequence

Crossrefs

Cf. A194833, A194834, A054065.

Programs

Mathematica

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 *)

Extensions

Table in overview corrected by Georg Fischer, Jul 30 2023