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 1-4 of 4 results.

A194836 Rectangular array, by antidiagonals: row n gives the positions of n in the fractal sequence A194835; an interspersion.

Original entry on oeis.org

1, 3, 2, 5, 4, 6, 9, 7, 10, 8, 13, 11, 14, 12, 15, 19, 16, 20, 17, 21, 18, 26, 23, 27, 24, 28, 25, 22, 33, 30, 35, 31, 36, 32, 29, 34, 42, 38, 44, 40, 45, 41, 37, 43, 39, 51, 47, 53, 49, 55, 50, 46, 52, 48, 54, 62, 57, 64, 59, 66, 61, 56, 63, 58, 65, 60, 74, 69, 76
Offset: 1

Views

Author

Clark Kimberling, Sep 03 2011

Keywords

Comments

Each pair of rows eventually intersperse.

Examples

			 Northwest corner:
1...3...5...9...13..19..26
2...4...7...11..16..23..30
6...10..14..20..27..35..44
8...12..17..24..31..40..49
15..21..28..36..45..55..66
18..25..32..41..50..61..73

Crossrefs

Cf. A194835, A194837, A194832.

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}]]  (* A194835 *)
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}]] (* A194836 *)
q[n_] := Position[p, n]; Flatten[Table[q[n],
 {n, 1, 80}]]  (* A194837 *)

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

A194837 Inverse permutation of A194836; every positive integer occurs exactly once.

Original entry on oeis.org

1, 3, 2, 5, 4, 6, 8, 10, 7, 9, 12, 14, 11, 13, 15, 17, 19, 21, 16, 18, 20, 28, 23, 25, 27, 22, 24, 26, 35, 30, 32, 34, 29, 36, 31, 33, 43, 38, 45, 40, 42, 37, 44, 39, 41, 52, 47, 54, 49, 51, 46, 53, 48, 55, 50, 62, 57, 64, 59, 66, 61, 56, 63, 58, 65, 60, 78, 73, 68
Offset: 1

Views

Author

Clark Kimberling, Sep 03 2011

Keywords

Crossrefs

Cf. A194836, A194835, A194832.

Programs

A257811 Circle of fifths cycle (clockwise).

Original entry on oeis.org

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

Views

Author

Peter Woodward, May 09 2015

Keywords

Comments

The twelve notes dividing the octave are numbered 1 through 12 sequentially. This sequence begins at a certain note, travels up a perfect fifth (seven semitones) twelve times, and arrives back at the same note. If justly tuned fifths are used, the final note will be sharp by the Pythagorean comma (roughly 23.46 cents or about a quarter of a semitone).
Period 12: repeat [1, 8, 3, 10, 5, 12, 7, 2, 9, 4, 11, 6]. - Omar E. Pol, May 12 2015

Examples

			For a(3), 1+7+7 == 3 (mod 12).
For a(4), 1+7+7+7 == 10 (mod 12).

Links

Crossrefs

Cf. A194835 (Contains this circle of fifths sequence), A007337 (sqrt(3) sequence), A258054 (counterclockwise circle of fifths cycle).

Programs

  • Magma
    [1+7*(n-1) mod(12): n in [1..80]]; // Vincenzo Librandi, May 10 2015
  • Mathematica
    PadRight[{}, 100, {1, 8, 3, 10, 5, 12, 7, 2, 9, 4, 11, 6}] (* Vincenzo Librandi, May 10 2015 *)
LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1},{1, 8, 3, 10, 5, 12, 7, 2, 9, 4, 11, 6},108] (* Ray Chandler, Aug 27 2015 *)
  • PARI
    a(n)=7*(n-1)%12+1 \\ Charles R Greathouse IV, Jun 02 2015
  • PARI
    Vec(x*(1 + 8*x + 3*x^2 + 10*x^3 + 5*x^4 + 12*x^5 + 7*x^6 + 2*x^7 + 9*x^8 + 4*x^9 + 11*x^10 + 6*x^11) / (1 - x^12) + O(x^80)) \\ Colin Barker, Nov 15 2019

Formula

Periodic with period 12: a(n) = 1 + 7*(n-1) mod 12.
From Colin Barker, Nov 15 2019: (Start)
G.f.: x*(1 + 8*x + 3*x^2 + 10*x^3 + 5*x^4 + 12*x^5 + 7*x^6 + 2*x^7 + 9*x^8 + 4*x^9 + 11*x^10 + 6*x^11) / (1 - x^12).
a(n) = a(n-12) for n > 12.
(End)

Extensions

Extended by Ray Chandler, Aug 27 2015
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