A194832 -id:A194832 - cp's OEIS frontend

A194833 Rectangular array, by antidiagonals: row n gives the positions of n in the fractal sequence A194832; an interspersion.

1, 2, 3, 5, 6, 4, 8, 10, 7, 9, 12, 14, 11, 13, 15, 18, 20, 16, 19, 21, 17, 24, 27, 22, 25, 28, 23, 26, 32, 35, 30, 33, 36, 31, 34, 29, 40, 44, 38, 42, 45, 39, 43, 37, 41, 49, 53, 47, 51, 55, 48, 52, 46, 50, 54, 60, 64, 57, 62, 66, 59, 63, 56, 61, 65, 58, 71, 76, 68

Offset: 1

Clark Kimberling, Sep 03 2011

As a sequence, A194833 is a permutation of the positive integers; its inverse is A194834.

			Northwest corner:
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

Cf. A194832, A194834.

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 *)

A374411 Triangle T(n, k) read by rows: Maximum number of linear patterns of length k in a circular permutation of length n taken from row n in A194832.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 6, 4, 1, 2, 6, 16, 5, 1, 2, 6, 20, 25, 6, 1, 2, 6, 24, 60, 36, 7, 1, 2, 6, 24, 85, 126, 49, 8, 1, 2, 6, 24, 100, 222, 196, 64, 9, 1, 2, 6, 24, 115, 390, 511, 288, 81, 10, 1, 2, 6, 24, 120, 558, 1085, 912, 405, 100, 11, 1, 2, 6, 24, 120, 654, 1911, 2328, 1458, 550, 121, 12

Offset: 1

Thomas Scheuerle, Jul 08 2024

Pattern counting considers only one revolution otherwise every sufficiently long circular permutation, with enough revolutions allowed, contains every pattern.

Each column k is divisible by k, because as we count linear patterns inside a circular permutation, we may obtain all circular shifts of the subset which represents a particular pattern.

			The triangle begins:
   n| k: 1| 2| 3|  4|   5|   6|   7|  8|  9
  =========================================
  [1]    1
  [2]    1, 2
  [3]    1, 2, 3
  [4]    1, 2, 6,  4
  [5]    1, 2, 6, 16,   5
  [6]    1, 2, 6, 20,  25,   6
  [7]    1, 2, 6, 24,  60,  36,   7
  [8]    1, 2, 6, 24,  85, 126,  49,  8
  [9]    1, 2, 6, 24, 100, 222, 196, 64, 9
.
Row 5 of A194832 is [3, 1, 4, 2, 5].
T(5, 4) = 16 because we will find these 16 distinct patterns of length 4:
   [3, 1, 4, 2] [1, 4, 2, 3] [4, 2, 3, 1] [2, 3, 1, 4]
 These are rotations of the ordering [1, 4, 2, 3].
   [1, 4, 2, 5] [4, 2, 5, 1] [2, 5, 1, 4] [5, 1, 4, 2]
 These are rotations of the ordering [1, 3, 2, 4].
   [2, 5, 3, 1] [5, 3, 1, 2] [3, 1, 2, 5] [1, 2, 5, 3]
 These are rotations of the ordering [1, 2, 4, 3].
   [5, 3, 1, 4] [3, 1, 4, 5] [1, 4, 5, 3] [4, 5, 3, 1]
 These are rotations of the ordering [1, 3, 4, 2].

Cf. A194832, A371823, A373778.

T(n, k+1)/(k+1) <= A371823(n-1, k) <= A373778(n-1, k).

A054065 Fractal sequence induced by tau: for k >= 1, let p(k) be the permutation of 1,2,...,k obtained by ordering the fractional parts {h*tau} for h=1,2,...,k; then juxtapose p(1),p(2),p(3),...

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

nonn

			p(1)=(1); p(2)=(2,1); p(3)=(2,1,3); p(4)=(2,4,1,3).
As a triangular array (see A194832), first nine rows:
1
2 1
2 1 3
2 4 1 3
5 2 4 1 3
5 2 4 1 6 3
5 2 7 4 1 6 3
5 2 7 4 1 6 3 8
5 2 7 4 9 1 6 3 8

Position of 1 in p(k) is given by A019446. Position of k in p(k) is given by A019587. For related arrays and sequences, see A194832.

r = (1 + Sqrt[5])/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}]] (* A054065 *)
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}]] (* A054069 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 80}]]  (* A054068 *)
(* Clark Kimberling, Sep 03 2011 *)
Flatten[Table[Ordering[Table[FractionalPart[GoldenRatio k], {k, n}]], {n, 10}]] (* Birkas Gyorgy, Jun 30 2012 *)

Extended by Ray Chandler, Apr 18 2009

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

Clark Kimberling

nonn

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

			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

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A054071, A054072, A194832.

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 *)

A194905 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=Pi.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 05 2011

See A194832 for a general discussion.

			First nine rows:
1
1 2
1 2 3
1 2 3 4
1 2 3 4 5
1 2 3 4 5 6
1 2 3 4 5 6 7
8 1 2 3 4 5 6 7
8 1 9 2 3 4 5 6 7

Cf. A194832, A194906, A194907.

r = Pi;
t[n_] := Table[FractionalPart[k*r], {k, 1, n}];
f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@
Sort[t[n], Less]], {n, 1, 20}]]  (* A194905 *)
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}]]  (* A194906 *)
q[n_] := Position[p, n]; Flatten[Table[q[n],
{n, 1, 80}]]  (* A194907 *)

A194835 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=-sqrt(2).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 03 2011

See A194832 for a general discussion.

			First nine rows:
1
2 1
2 1 3
2 4 1 3
2 4 1 3 5
2 4 6 1 3 5
7 2 4 6 1 3 5
7 2 4 6 1 8 3 5
7 2 9 4 6 1 8 3 5

Cf. A194832, A194836, A194837.

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 *)

A194841 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=-sqrt(3).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 04 2011

See A194832 for a general discussion.

			First nine rows:
1
1 2
1 2 3
4 1 2 3
4 1 5 2 3
4 1 5 2 6 3
4 1 5 2 6 3 7
4 8 1 5 2 6 3 7
4 8 1 5 9 2 6 3 7

Cf. A194832, A194842, A194843.

r = -Sqrt[3];
t[n_] := Table[FractionalPart[k*r], {k, 1, n}];
f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@ Sort[t[n], Less]], {n, 1,20}]]   (* A194841 *)
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}]]    (* A194842 *)
q[n_] := Position[p, n]; Flatten[
 Table[q[n], {n, 1, 80}]]   (* A194843 *)

A194842 Rectangular array, by antidiagonals: row n gives the positions of n in the fractal sequence A194841; an interspersion.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 7, 12, 14, 15, 11, 13, 17, 19, 21, 16, 18, 20, 23, 25, 27, 22, 24, 26, 28, 31, 33, 35, 29, 32, 34, 36, 30, 39, 42, 44, 37, 40, 43, 45, 38, 41, 48, 51, 54, 46, 49, 52, 55, 47, 50, 53, 58, 61, 64, 56, 59, 62, 65, 57, 60, 63, 66, 70, 73, 76

Offset: 1

Clark Kimberling, Sep 04 2011

See A194832 and A194841.

			Northwest corner:
1...2...4...8...12..17..23
3...5...9...14..19..25..33
6...10..15..21..27..35..44
7...11..16..22..29..37..46
13..18..24..32..40..49..59

Cf. A194841, A194843, A194832.

r = -Sqrt[3];
t[n_] := Table[FractionalPart[k*r], {k, 1, n}];
f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@ Sort[t[n], Less]], {n, 1,20}]]   (* A194841 *)
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}]]    (* A194842 *)
q[n_] := Position[p, n]; Flatten[
 Table[q[n], {n, 1, 80}]]   (* A194843 *)

A194868 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=-(1+sqrt(3))/2.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 04 2011

See A194832 for a general discussion.

			First nine rows:
1
2 1
2 1 3
2 4 1 3
5 2 4 1 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 1 9 6 3

Cf. A194832, A194869, A194870.

r = -(1 + Sqrt[3])/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}]]  (* A194868 *)
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}]] (* A194869 *)
q[n_] := Position[p, n]; Flatten[Table[q[n],
{n, 1, 80}]]   (* A194870 *)

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

Clark Kimberling, Sep 03 2011

Each pair of rows eventually intersperse.

			 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

Cf. A194835, A194837, A194832.

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 *)

cp's OEIS Frontend

A194833 Rectangular array, by antidiagonals: row n gives the positions of n in the fractal sequence A194832; an interspersion.

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A374411 Triangle T(n, k) read by rows: Maximum number of linear patterns of length k in a circular permutation of length n taken from row n in A194832.

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Formula

A054065 Fractal sequence induced by tau: for k >= 1, let p(k) be the permutation of 1,2,...,k obtained by ordering the fractional parts {h*tau} for h=1,2,...,k; then juxtapose p(1),p(2),p(3),...

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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),...

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A194905 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=Pi.

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A194835 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=-sqrt(2).

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A194841 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=-sqrt(3).

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A194842 Rectangular array, by antidiagonals: row n gives the positions of n in the fractal sequence A194841; an interspersion.

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A194868 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=-(1+sqrt(3))/2.

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