A054065 -id:A054065 - cp's OEIS frontend

A054068 Permutation of N = set of natural numbers: a(n)+C(k,2), where a=A054065 and k=Floor((1+sqrt(8n-3))/2).

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

Offset: 1

Clark Kimberling

nonn

			Referring to A054065, just add C(k,2) to the numbers in p(k): e.g. p(1)=(1)->(1); p(2)=(2,1)->(3,2); p(3)=(2,1,3)->(5,4,6).

Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A054069.

A100056 Sum of absolute differences of p(n) defined in A054065.

Original entry on oeis.org

1, 3, 7, 10, 16, 22, 27, 37, 47, 57, 67, 75, 91, 107, 123, 139, 155, 171, 187, 200, 226, 252, 278, 304, 330, 356, 382, 408, 434, 460, 486, 512, 533, 575, 617, 659, 701, 743, 785, 827, 869, 911, 953, 995, 1037, 1079, 1121, 1163, 1205, 1247, 1289, 1331, 1373

Offset: 2

Gary W. Adamson, Oct 31 2004

nonn

Conjecture: a(n) provides the maximum sum of difference terms of any permutation of 1 through n. A054065 (or reversals) give the optimal permutations and a(n) gives the sum.

			a(4) = 7 since p(4) of A054065 = 2 4 1 3. Finite difference row (abs. values) of p(4) = 2 3 2; sum = 7.

Cf. A100057, clock-oriented analogous sequence.

a(n) = sum, (abs. values) of difference row of p(n), A054065.

Edited and extended by Ray Chandler, Apr 18 2009

A100057 Sum of absolute differences of p(n) defined in A054065, oriented around a clock.

Original entry on oeis.org

2, 4, 8, 12, 18, 24, 30, 40, 50, 60, 70, 80, 96, 112, 128, 144, 160, 176, 192, 208, 234, 260, 286, 312, 338, 364, 390, 416, 442, 468, 494, 520, 546, 588, 630, 672, 714, 756, 798, 840, 882, 924, 966, 1008, 1050, 1092, 1134, 1176, 1218, 1260, 1302, 1344, 1386

Offset: 2

Gary W. Adamson, Oct 31 2004

nonn

			a(4) = 8 since p(4) of A054065 = 2 4 1 3. Placing the terms in circular fashion and taking absolute differences: 2, 3, 2, 1; sum = 8.

Cf. A100056.

Arrange p(n) of A054065 around a clock. a(n) = sum of absolute differences between terms. Alternatively, take the absolute difference between left and rightmost terms of p(n), A054065; and add to A100056(n).

Edited and extended by Ray Chandler, Apr 18 2009

A132283 Normalization of dense fractal sequence A054065 (defined from fractional parts {n*tau}, where tau = golden ratio).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Aug 16 2007

nonn

A fractal sequence, dense in the sense that if i,j are neighbors in a segment, then eventually i and j are separated by some k in all later segments. (Hence in the "limit", i,j are separated by infinitely many other numbers.)

			Start with A054065=(1,2,1,2,1,3,2,4,1,3,5,2,4,1,3,5,2,4,1,6,3,5,2,...)
Step 1. Append initial 1.
Step 2. Write segments: 1; 1,2; 1,2; 1,3,2,4; 1,3,5,2,4;...
Step 3. Delete repeated segments: 1; 1,2; 1,3,2,4; 1,3,5,2,4; ...
Step 4. Make segment #n have length n by allowing only newcomer, namely n, like this: 1; 1,2; 1,3,2; 1,3,2,4; 1,3,5,2,4; 1,6,3,5,2,4; ...
Step 5. Concatenate those segments.

C. Kimberling, Proper self-containing sequences, fractal sequences and para-sequences, preprint, 2007.

Clark Kimberling, Self-Containing Sequences, Selection Functions, and Parasequences, J. Int. Seq. Vol. 25 (2022), Article 22.2.1.

Cf. A132284.

A054066 Position of n-th 1 in A054065.

Original entry on oeis.org

1, 3, 5, 9, 14, 19, 26, 33, 42, 52, 62, 74, 87, 100, 115, 130, 147, 165, 183, 203, 223, 245, 268, 291, 316, 342, 368, 396, 424, 454, 485, 516, 549, 583, 617, 653, 689, 727, 766, 805, 846, 887, 930, 974, 1018, 1064, 1111, 1158, 1207

Offset: 1

Clark Kimberling

nonn

A054067 Position of first appearance of n in A054065.

Original entry on oeis.org

1, 2, 6, 8, 11, 20, 24, 36, 41, 47, 64, 71, 79, 101, 110, 135, 145, 156, 186, 198, 231, 244, 258, 296, 311, 327, 370, 387, 433, 451, 470, 521, 541, 562, 618, 640, 699, 722, 746, 810, 835, 902, 928, 955, 1027, 1055, 1084, 1161, 1191

Offset: 1

Clark Kimberling

nonn

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

Clark Kimberling, Sep 03 2011

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

			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

C. Kimberling, Fractal sequences and interspersions, Ars Combinatoria 45 (1997), 157-168.

Wikipedia, Fractal sequence

Cf. A194833, A194834, A054065.

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 *)

Table in overview corrected by Georg Fischer, Jul 30 2023

A054069 Inverse of the permutation A054068 of natural numbers.

Original entry on oeis.org

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.)
```

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 */
```

A133117 Fractal sequence based on comparison of {n * tau} with {itau} for i = 1 to F(2j) where F(2j) equals the first i for which {ntau} <= {itau} as i goes from 1 to F(2j+2)-1 and F(2j) equals the insertion point of n into P(n-1). The fractional parts {itau} are all less than or equal to {F(2j-2)*tau} for 0 < i < F(2j), so there is no chance that an insertion point greater than n in the permutation of the first n-1 integers will be specified by this rule. The table, A132827, gives the insertion points for each n into the permutation P(n-1) of the first n integers.

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 4, 2, 1, 3, 5, 4, 2, 1, 3, 5, 4, 6, 2, 1, 3, 7, 5, 4, 6, 2, 1, 3, 7, 5, 4, 6, 2, 1, 3, 8

Offset: 1

Kenneth J Ramsey, Sep 13 2007

This sequence is a modification of that in A054065 which gives the fractal series of the same permutation as the permutation of A132917 for which a couple of generating algorithms are given.

			The first few permutations are 1, 21, 213, 4213, 54213, 546213 since {6*tau} is greater than {1*Tau} but less than {3*Tau}; and since of 0<i<7 only {3*tau} and {6*tau} are greater than {1*tau}

Cf. A054065, A132827, A132917, A132828.

See A132827.

cp's OEIS Frontend

A054068 Permutation of N = set of natural numbers: a(n)+C(k,2), where a=A054065 and k=Floor((1+sqrt(8n-3))/2).

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A100056 Sum of absolute differences of p(n) defined in A054065.

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A100057 Sum of absolute differences of p(n) defined in A054065, oriented around a clock.

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A132283 Normalization of dense fractal sequence A054065 (defined from fractional parts {n*tau}, where tau = golden ratio).

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A054066 Position of n-th 1 in A054065.

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A054067 Position of first appearance of n in A054065.

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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.

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A054069 Inverse of the permutation A054068 of natural numbers.

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

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