A194832 -id:A194832 - cp's OEIS frontend

A054069 Inverse of the permutation A054068 of natural numbers.

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, 33, 30, 35, 32, 29, 34, 31, 36, 42, 38, 44, 40, 37, 43, 39, 45, 41, 52, 48, 54, 50, 46, 53, 49, 55, 51, 47, 62, 58, 65, 60, 56, 63, 59, 66

Offset: 1

Clark Kimberling

nonn

As an interspersion, row n gives the positions of n in the fractal sequence A054065.

			Northwest corner, as an interspersion:
1...2...5...8...12..18..24
3...6...10..14..20..27..35
4...7...11..16..22..30..38
9...13..19..25..33..42..51
15..21..28..36..45..55..66

Index entries for sequences that are permutations of the natural numbers

A054065, A054068, A194832.

Mathematica
```
(See A054065.)
```

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

Clark Kimberling, Sep 03 2011

nonn

Cf. A194836, A194835, A194832.

Mathematica
```
(See A194835.)
```

A194840 Inverse permutation of A194839; every positive integer occurs exactly once.

Original entry on oeis.org

1, 3, 2, 6, 5, 4, 9, 8, 7, 10, 13, 12, 15, 11, 14, 18, 21, 17, 20, 16, 19, 28, 24, 27, 23, 26, 22, 25, 35, 31, 34, 30, 33, 29, 36, 32, 43, 39, 42, 38, 45, 41, 37, 44, 40, 52, 48, 55, 51, 47, 54, 50, 46, 53, 49, 66, 62, 58, 65, 61, 57, 64, 60, 56, 63, 59, 77, 73, 69

Offset: 1

Clark Kimberling, Sep 03 2011

nonn

Cf. A194838, A194839, A194832.

Mathematica
```
(See A194839.)
```

A194864 Inverse permutation of A194866; every positive integer occurs exactly once.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 10, 7, 8, 9, 14, 11, 15, 12, 13, 19, 16, 20, 17, 21, 18, 25, 22, 26, 23, 27, 24, 28, 32, 36, 29, 33, 30, 34, 31, 35, 40, 44, 37, 41, 45, 38, 42, 39, 43, 49, 53, 46, 50, 54, 47, 51, 55, 48, 52, 66, 59, 63, 56, 60, 64, 57, 61, 65, 58, 62, 77, 70, 74

Offset: 1

Clark Kimberling, Sep 04 2011

nonn

Cf. A194832, A194865, A194866.

Mathematica
```
(See A194865.)
```

Definition corrected by Georg Fischer, Jul 30 2023

A194867 Inverse permutation of A194863; every positive integer occurs exactly once.

Original entry on oeis.org

1, 2, 3, 6, 4, 5, 9, 7, 10, 8, 13, 11, 14, 12, 15, 18, 21, 16, 19, 17, 20, 24, 27, 22, 25, 28, 23, 26, 31, 34, 29, 32, 35, 30, 33, 36, 39, 42, 45, 37, 40, 43, 38, 41, 44, 48, 51, 54, 46, 49, 52, 55, 47, 50, 53, 66, 58, 61, 64, 56, 59, 62, 65, 57, 60, 63, 77, 69, 72

Offset: 1

Clark Kimberling, Sep 04 2011

nonn

Cf. A194832, A194862, A194863.

Mathematica
```
(See A194862.)
```

Definition corrected by Georg Fischer, Jul 30 2023

A194834 Inverse permutation of A194833; every positive integer occurs exactly once.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 03 2011

nonn

Cf. A194832, A194833.

Mathematica
```
(See A194832.)
```

A215345 Value of y in the n-th number of the form x+y*(1+sqrt(5))/2.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 2, 1, 0, 2, 1, 3, 0, 2, 1, 3, 0, 2, 4, 1, 3, 0, 2, 4, 1, 3, 0, 5, 2, 4, 1, 3, 0, 5, 2, 4, 1, 6, 3, 0, 5, 2, 4, 1, 6, 3, 0, 5, 2, 7, 4, 1, 6, 3, 0, 5, 2, 7, 4, 1, 6, 3, 8, 0, 5, 2, 7, 4, 1, 6, 3, 8, 0, 5, 2, 7, 4, 9, 1, 6, 3, 8, 0, 5, 2, 7, 4, 9, 1, 6, 3, 8, 0, 5, 10, 2, 7, 4, 9, 1, 6, 3, 8, 0, 5, 10, 2, 7, 4, 9, 1, 6, 11, 3, 8

Offset: 1

Peter G. Anderson, Aug 08 2012

nonn

Let x>=0, y>=0 be integers, sort according to x+y*(1+sqrt(5))/2, this sequence gives the y-values. - Joerg Arndt, Aug 16 2012

			Let g = (1+sqrt(5))/2, sequences A215344 (x) and A215345 (y) start as:
[x+y*g, x, y]
[0.0000000, 0, 0]
[1.0000000, 1, 0]
[1.6180340, 0, 1]
[2.0000000, 2, 0]
[2.6180340, 1, 1]
[3.0000000, 3, 0]
[3.2360680, 0, 2]
[3.6180340, 2, 1]
[4.0000000, 4, 0]
[4.2360680, 1, 2]
[4.6180340, 3, 1]
[4.8541020, 0, 3]
[5.0000000, 5, 0]
[5.2360680, 2, 2]
[5.6180340, 4, 1]
- _Joerg Arndt_, Aug 17 2012.

Peter G. Anderson, Table of n, a(n) for n = 1..1571

A215344 is the value of x.

Cf. A084531, A054065, A194832.

PARI
```
/* see A215344 */
```

A372231 Fixed points of A372341.

Original entry on oeis.org

1, 2, 5, 8, 12, 18, 24, 32, 40, 49, 60, 71, 83, 97, 111, 127, 143, 160, 179, 198, 219, 240, 262, 286, 310, 335, 362, 389, 418, 447, 477, 509, 541, 574, 609, 644, 681, 718, 756, 796, 836, 878, 920, 963, 1008, 1053, 1099, 1147, 1195, 1245, 1295, 1346, 1399, 1452

Offset: 1

Rémy Sigrist, Apr 28 2024

nonn

			A372341(49) = 49, so 49 belongs to this sequence.

Rémy Sigrist, C++ program

Cf. A000217, A005206, A194832, A194833, A372341.

a(n) = A000217(n) - A005206(n-1).

A194832(a(n)) = 1. Also the first row A194833(1, n) = a(n). This can be seen in the EXAMPLE section of A372341. Each row is horizontally shifted by a value from A005206. The shift is known to be floor((k+1)*tau)-k-1, where tau is the golden ratio. The position of "1" in a permutation from A194832 is determined by a similar process based on the same constant. - Thomas Scheuerle, Jul 08 2024

A371822 Triangle read by rows. Row n is the lexicographically earliest permutation of [n] that can be obtained from row n-1 by inserting the element n and optional cyclic shifting to maximize the pattern density.

Original entry on oeis.org

1, 1, 2, 3, 1, 2, 3, 1, 4, 2, 2, 5, 3, 1, 4, 2, 5, 3, 6, 1, 4, 3, 6, 1, 4, 7, 2, 5, 2, 5, 8, 3, 6, 1, 4, 7, 6, 1, 9, 4, 7, 2, 5, 8, 3, 7, 2, 10, 5, 8, 3, 6, 1, 9, 4, 7, 2, 10, 5, 8, 3, 11, 6, 1, 9, 4, 8, 3, 11, 6, 1, 9, 4, 12, 7, 2, 10, 5, 11, 6, 1, 9, 4, 12, 7, 2, 10, 5, 13, 8, 3, 11, 6, 1, 14, 9, 4, 12, 7, 2, 10, 5, 13, 8, 3

Offset: 1

Thomas Scheuerle, Jun 22 2024

The first 13 rows include shortest k-superpatterns for k up to 5. These k-superpatterns are also optimal superpatterns. Optimal means the overall pattern density including patterns of all length is maximal among all permutations of [n]. How many more shortest superpatterns will be given by this sequence? The next will be expected in row 17.
Row n is a k-superpattern if row n of A371823 starts with 1!, 2!, ..., k!. If n also coincides with A342474(k), then row n is a shortest possible k-superpattern.
At time of sequence publication, all known rows agree up to cyclic shift with rows from A194832. This could indicate that A194832 will at least almost optimize the pattern density for permutations on the circle.
The above observations are accompanied by strong statistical arguments: because (1+sqrt(5))/2 has the simplest continued fraction expansion of any irrational number it will optimize asymptotically the pattern density in the permutations induced by it.

			The first 10 rows:
  1
  1,  2
  3,  1,  2
  3,  1,  4,  2
  2,  5,  3,  1,  4
  2,  5,  3,  6,  1,  4
  3,  6,  1,  4,  7,  2,  5
  2,  5,  8,  3,  6,  1,  4,  7
  6,  1,  9,  4,  7,  2,  5,  8,  3
  7,  2, 10,  5,  8,  3,  6,  1,  9,  4

Wikipedia, Superpattern.

A371823 lists the number of different patterns of length k in row n.
Cf. A194832 (same rows cyclically shifted?).
Cf. A342474, A373778.

Edited by Peter Munn, Jul 09 2024

cp's OEIS Frontend

A054069 Inverse of the permutation A054068 of natural numbers.

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

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A194840 Inverse permutation of A194839; every positive integer occurs exactly once.

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A194864 Inverse permutation of A194866; every positive integer occurs exactly once.

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A194867 Inverse permutation of A194863; every positive integer occurs exactly once.

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A194834 Inverse permutation of A194833; every positive integer occurs exactly once.

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Mathematica

A215345 Value of y in the n-th number of the form x+y*(1+sqrt(5))/2.

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PARI

A372231 Fixed points of A372341.

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Formula

A371822 Triangle read by rows. Row n is the lexicographically earliest permutation of [n] that can be obtained from row n-1 by inserting the element n and optional cyclic shifting to maximize the pattern density.

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