A194868 -id:A194868 - cp's OEIS frontend

A194869 Rectangular array, by antidiagonals: row n gives the positions of n in the fractal sequence A194868; an interspersion.

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

Offset: 1

Clark Kimberling, Sep 04 2011

Every pair of rows eventually intersperse.

			Northwest corner:
1...3...5...9...14..19
2...4...7...12..17..23
6...10..15..21..28..36
8...13..18..25..33..41
11..16..22..30..38..47
20..27..35..44..54..64

Cf. A194832, A194868, A194870.

r = -(1 + Sqrt[3])/2;
t[n_] := Table[FractionalPart[k*r], {k, 1, n}];
f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@
Sort[t[n], Less]], {n, 1, 20}]]  (* A194868 *)
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}]] (* A194869 *)
q[n_] := Position[p, n]; Flatten[Table[q[n],
{n, 1, 80}]]   (* A194870 *)

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

A194870 Inverse permutation of A194869; every positive integer occurs exactly once.

Original entry on oeis.org

1, 3, 2, 5, 4, 6, 8, 10, 7, 9, 15, 12, 14, 11, 13, 20, 17, 19, 16, 21, 18, 26, 23, 28, 25, 22, 27, 24, 36, 33, 30, 35, 32, 29, 34, 31, 44, 41, 38, 43, 40, 37, 45, 42, 39, 53, 50, 47, 55, 52, 49, 46, 54, 51, 48, 63, 60, 57, 65, 62, 59, 56, 64, 61, 58, 66, 74, 71, 68

Offset: 1

Clark Kimberling, Sep 04 2011

nonn

Cf. A194868, A194869.

Mathematica
```
(See A194868.)
```

A376448 a(n) = k if k is odd otherwise a(n) = k+1 and k = floor( 2^n*(1+sqrt(5))/2 ).

Original entry on oeis.org

1, 3, 7, 13, 25, 51, 103, 207, 415, 829, 1657, 3313, 6627, 13255, 26509, 53019, 106039, 212079, 424157, 848315, 1696631, 3393263, 6786527, 13573053, 27146105, 54292211, 108584423, 217168845, 434337691, 868675383, 1737350767, 3474701533, 6949403065, 13898806131, 27797612261, 55595224523

Offset: 0

Thomas Scheuerle, Sep 23 2024

The sequence of all multiples of an irrational b is equidistributed modulo 1. Such a sequence is called a Weyl sequence. It is common practice in computing to approximate a Weyl sequence by taking integer multiples of some integer m modulo a power of two. This requires that the integer m is odd. This sequence provides suitable m = a(n) for the case modulo 2^n. It utilizes the golden ratio for approximation of irrationality.

			An example for a pseudo Weyl sequence obtained from a(3):
{0, 1, 2, 3, 4, 5, 6, 7} * a(3) mod 2^3 = {0, 5, 2, 7, 4, 1, 6, 3}. (Without zero also part of A194868).

Cf. A054065, A194868, A293313, A293314, A293315.

k[n_]:=Floor[2^n*GoldenRatio];Table[If[OddQ[k[n]],k[n],k[n]+1],{n,0,35}] (* James C. McMahon, Oct 20 2024 *)

PARI
```
a(n) = {my( m=floor(quadgen(5)<
				
```

a(n) = 2*A293313(n-1) + 1, for n > 0.

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A194869 Rectangular array, by antidiagonals: row n gives the positions of n in the fractal sequence A194868; an interspersion.

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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.

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A194870 Inverse permutation of A194869; every positive integer occurs exactly once.

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A376448 a(n) = k if k is odd otherwise a(n) = k+1 and k = floor( 2^n*(1+sqrt(5))/2 ).

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