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.

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A054078 Number of k<=n such that a(k)>n, where a is the permutation A054076 of N.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 1, 2, 1, 0, 1, 2, 1, 2, 1, 0, 0, 1, 2, 2, 2, 2, 1, 1, 0, 1, 2, 2, 2, 2, 1, 2, 1, 0, 1, 2, 2, 3, 2, 2, 2, 2, 1, 0, 1, 2, 2, 3, 2, 2, 3, 2, 2, 1, 0, 1, 2, 2, 3, 2, 2, 3, 2, 2, 1, 0, 0, 1, 2, 2, 3, 3, 3, 3, 3, 3, 2, 1, 1
Offset: 1

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Keywords

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

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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.

Crossrefs

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

A054073 Fractal sequence induced by sqrt(2): for k >= 1 let p(k) be the permutation of 1,2,...,k obtained by ordering the fractional parts {h*sqrt(2)} for h=1,2,...,k; then juxtapose p(1),p(2),p(3),...

Original entry on oeis.org

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

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Keywords

Comments

A054073 generates the interspersion A054077; see A194832 and the Mathematica program.

Examples

			p(1)=(1); p(2)=(1,2); p(3)=(3,1,2); p(4)=(3,1,4,2).
When formatted as a triangle, the first 9 rows:
1
1 2
3 1 2
3 1 4 2
5 3 1 4 2
5 3 1 6 4 2
5 3 1 6 4 2 7
5 3 8 1 6 4 2 7
5 3 8 1 6 4 9 2 7
		

Crossrefs

Programs

  • Mathematica
    r = Sqrt[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}]] (* A054073 *)
    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}]] (* A054077 *)
    q[n_] := Position[p, n]; Flatten[
    Table[q[n], {n, 1, 80}]]  (* A054076 *)
    (* Clark Kimberling, Sep 03 2011 *)

A054077 Inverse of the permutation A054073 of N.

Original entry on oeis.org

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

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Keywords

Comments

When formatted as a rectangular array, row n gives the positions of n in A054073; the array is an interspersion. (Each pair of rows eventually intersperse.)

Examples

			Northwest corner when formatted as an interspersion:
1...2...5...8...13..18
3...6...10..15..21..27
4...7...12..17..23..30
9...14..20..26..34..42
11..16..22..29..37..46
19..25..33..41..51..61
		

Programs

  • Mathematica
    r = Sqrt[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}]] (* A054073 *)
    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}]] (* A054077 *)
    q[n_] := Position[p, n]; Flatten[
     Table[q[n], {n, 1, 80}]]  (* A054076 *)
Showing 1-4 of 4 results.