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 1-7 of 7 results.

A054076 Permutation of N: a(n)+C(k,2), where a=A054073 and k=Floor((1+sqrt(8n-3))/2).

Original entry on oeis.org

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

Views

Author

Clark Kimberling

Keywords

Examples

			Referring to A054073, just add C(k,2) to the numbers in p(k); e.g. p(1)=1->(1); p(2)=(1,2)->(2,3); p(3)=(3,1,2)->(6,4,5).

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

Views

Author

Clark Kimberling

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 *)

A054074 Position of n-th 1 in A054073.

Original entry on oeis.org

1, 2, 5, 8, 13, 18, 24, 32, 40, 50, 60, 71, 84, 97, 112, 127, 144, 161, 179, 199, 219, 241, 263, 286, 311, 336, 363, 390, 419, 448, 478, 510, 542, 576, 610, 645, 682, 719, 758, 797, 837, 879, 921, 965, 1009, 1055, 1101, 1148, 1197
Offset: 1

Views

Author

Clark Kimberling

Keywords

A054075 Position of first appearance of n in A054073.

Original entry on oeis.org

1, 3, 4, 9, 11, 19, 28, 31, 43, 47, 62, 78, 83, 102, 108, 130, 137, 162, 188, 196, 225, 234, 266, 299, 309, 345, 356, 395, 407, 449, 492, 505, 551, 565, 614, 664, 679, 732, 748, 804, 861, 878, 938, 956, 1019, 1038, 1104, 1171, 1191
Offset: 1

Views

Author

Clark Kimberling

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

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

A258054 Circle of fifths cycle (counterclockwise).

Original entry on oeis.org

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

Views

Author

Peter Woodward, May 17 2015

Keywords

Comments

The twelve notes dividing the octave are numbered 1 through 12 sequentially. This sequence begins at a certain note, travels down a perfect fifth twelve times (seven semitones), and arrives back at the same note. If justly tuned fifths are used, the final note will be flat by the Pythagorean comma (roughly 23.46 cents or about a quarter of a semitone).
Period 12: repeat [1, 6, 11, 4, 9, 2, 7, 12, 5, 10, 3, 8]. - Omar E. Pol, May 18 2015
The string [1, 6, 11, 4, 9, 2, 7, 12, 5, 10, 3, 8] is also in both A023127 and A054073. - Omar E. Pol, May 19 2015

Examples

			For a(3), 1+5+5 = 11 (mod 12).
For a(4), 1+5+5+5 = 4 (mod 12).

Links

Crossrefs

Cf. A221363 (Pythagorean comma), A257811 (clockwise circle of fifths cycle).

Programs

  • Magma
    [1+5*(n-1) mod 12: n in [1..80]]; // Vincenzo Librandi, May 19 2015
  • Maple
    A258054:=n->1+(5*(n-1) mod 12): seq(A258054(n), n=1..100); # Wesley Ivan Hurt, May 22 2015
  • Mathematica
    PadRight[{}, 100, {1, 6, 11, 4, 9, 2, 7, 12, 5, 10, 3, 8}] (* Vincenzo Librandi, May 19 2015 *)
LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1},{1, 6, 11, 4, 9, 2, 7, 12, 5, 10, 3, 8},108] (* Ray Chandler, Aug 27 2015 *)
  • PARI
    a(n)=1+5*(n-1) \\ Charles R Greathouse IV, May 22 2015
  • PARI
    Vec(x*(1 + 6*x + 11*x^2 + 4*x^3 + 9*x^4 + 2*x^5 + 7*x^6 + 12*x^7 + 5*x^8 + 10*x^9 + 3*x^10 + 8*x^11) / (1 - x^12) + O(x^80)) \\ Colin Barker, Nov 15 2019

Formula

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

Extensions

Extended by Ray Chandler, Aug 27 2015

A173863 Square root of A172545(n).

Original entry on oeis.org

0, 1, 2, 0, 1, 2, 0, 3, 1, 2, 0, 3, 1, 4, 2, 0, 3, 1, 4, 2, 0, 3, 1, 4, 2, 0, 5, 3, 1, 4, 2, 0, 5, 3, 1, 4, 2, 0, 5, 3, 1, 6, 4, 2, 0, 5, 3, 1, 6, 4, 2, 0, 5, 3, 1, 6, 4, 2, 0, 5, 3, 1, 6, 4, 2, 0, 7, 5, 3, 1, 6, 4, 2, 0, 7, 5, 3, 1, 6, 4, 2, 0, 7, 5, 3, 1, 6, 4, 2, 0, 7, 5, 3, 1, 8, 6, 4, 2, 0, 7, 5, 3, 1, 8, 6, 4, 2, 0
Offset: 0

Views

Author

Paul Curtz, Nov 26 2010

Keywords

Crossrefs

A054073(n+2). Essentially A022337.
Showing 1-7 of 7 results.

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