cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A194857 Rectangular array, by antidiagonals: row n gives the positions of n in the fractal sequence A194856; an interspersion.

Original entry on oeis.org

1, 3, 2, 6, 5, 4, 10, 9, 8, 7, 14, 13, 12, 11, 15, 20, 18, 17, 16, 21, 19, 27, 25, 23, 22, 28, 26, 24, 35, 33, 31, 29, 36, 34, 32, 30, 43, 41, 39, 37, 44, 42, 40, 38, 45, 53, 50, 48, 46, 54, 51, 49, 47, 55, 52, 64, 61, 58, 56, 65, 62, 59, 57, 66, 63, 60, 76, 73, 70
Offset: 1

Views

Author

Clark Kimberling, Sep 04 2011

Keywords

Comments

See A194832 for a general discussion.

Examples

			Northwest corner:
1...3...6...10...14...20
2...5...9...13...18...25
4...8...12..17...23...51
7...11..16..22...29...37
15..21..38..36...44...54

Crossrefs

Cf. A194856, A194858, A194832.

Programs

  • Mathematica
    r = -Sqrt[5];
t[n_] := Table[FractionalPart[k*r], {k, 1, n}];
f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@
Sort[t[n], Less]], {n, 1, 20}]]  (* A194856 *)
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}]]  (* A194857 *)
q[n_] := Position[p, n]; Flatten[Table[q[n],
{n, 1, 80}]]  (* A194858 *)

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

Views

Author

Clark Kimberling, Sep 03 2011

Keywords

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

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

A194858 Inverse permutation to A194857; every positive integer occurs exactly once.

Original entry on oeis.org

1, 3, 2, 6, 5, 4, 10, 9, 8, 7, 14, 13, 12, 11, 15, 19, 18, 17, 21, 16, 20, 25, 24, 28, 23, 27, 22, 26, 32, 36, 31, 35, 30, 34, 29, 33, 40, 44, 39, 43, 38, 42, 37, 41, 45, 49, 53, 48, 52, 47, 51, 55, 46, 50, 54, 59, 63, 58, 62, 66, 57, 61, 65, 56, 60, 64, 70, 74, 78
Offset: 1

Views

Author

Clark Kimberling, Sep 04 2011

Keywords

Crossrefs

Cf. A194857, A194856.

Programs

Showing 1-3 of 3 results.

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