A194832 -id:A194832 - cp's OEIS frontend

A194897 Rectangular array, by antidiagonals: row n gives the positions of n in the fractal sequence A194896; an interspersion.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 16, 23, 25, 26, 27, 28, 22, 24, 30, 32, 34, 35, 36, 29, 31, 33, 38, 40, 42, 44, 45, 37, 39, 41, 43, 47, 49, 51, 53, 55, 46, 48, 50, 52, 54, 57, 59, 61, 63, 65, 56, 58, 60, 62, 64, 66, 69, 71, 73

Offset: 1

Clark Kimberling, Sep 04 2011

See A194832 for a general discussion.

			Northwest corner:
1...2...4...7...11..17..23
3...5...8...12..18..25..32
6...9...13..19..26..34..42
10..14..20..27..35..44..53
15..21..28..36..45..55..65
16..22..29..37..46..56..67

Cf. A194832, A194896, A194898.

r = -Sqrt[8];
t[n_] := Table[FractionalPart[k*r], {k, 1, n}];
f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@
Sort[t[n], Less]], {n, 1, 20}]] (* A194896 *)
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}]] (* A194897 *)
q[n_] := Position[p, n]; Flatten[Table[q[n],
{n, 1, 80}]] (* A194898 *)

A194899 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(12).

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

Offset: 1

Clark Kimberling, Sep 05 2011

See A194832 for a general discussion.

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

Cf. A194832, A194900, A194901.

r = Sqrt[12];
t[n_] := Table[FractionalPart[k*r], {k, 1, n}];
f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@
Sort[t[n], Less]], {n, 1, 20}]] (* A194899 *)
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, 15},
{k, 1, n}]] (* A194900 *)
q[n_] := Position[p, n]; Flatten[Table[q[n],
{n, 1, 90}]] (* A194901 *)

row(n) = Vec(vecsort(vector(n, k, frac(k*sqrt(12))),,1));
tabl(nn) = for (n=1, nn, print(row(n))); \\ Michel Marcus, Feb 06 2019

A194900 Rectangular array, by antidiagonals: row n gives the positions of n in the fractal sequence A194899; an interspersion.

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, 25, 28, 24, 27, 23, 26, 22, 32, 36, 31, 35, 30, 34, 29, 33, 41, 45, 40, 44, 39, 43, 38, 42, 37, 50, 55, 49, 54, 48, 53, 47, 52, 46, 51, 61, 66, 60, 65, 59, 64, 58, 63, 57, 62, 56, 72, 78, 71

Offset: 1

Clark Kimberling, Sep 05 2011

See A194832 for a general discussion.

			Northwest corner:
1...2...5...8...13..18
3...6...10..15..21..28
4...7...12..17..24..31
9...14..20..27..35..44
11..16..23..30..39..48

Cf. A194832, A194899, A194901.

r = Sqrt[12];
t[n_] := Table[FractionalPart[k*r], {k, 1, n}];
f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@
Sort[t[n], Less]], {n, 1, 20}]] (* A194899 *)
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, 15},
{k, 1, n}]] (* A194900 *)
q[n_] := Position[p, n]; Flatten[Table[q[n],
{n, 1, 90}]] (* A194901 *)

A194902 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(12).

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

Offset: 1

Clark Kimberling, Sep 05 2011

See A194832 for a general discussion.

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

Cf. A194832, A194903, A194904.

r = -Sqrt[12];
t[n_] := Table[FractionalPart[k*r], {k, 1, n}];
f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@
Sort[t[n], Less]], {n, 1, 20}]] (* A194902 *)
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}]] (* A194903 *)
q[n_] := Position[p, n]; Flatten[Table[q[n],
{n, 1, 80}]]  (* A194904 *)

A194903 Rectangular array, by antidiagonals: row n gives the positions of n in the fractal sequence A194902; 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, 25, 22, 26, 23, 27, 24, 28, 33, 29, 34, 30, 35, 31, 36, 32, 41, 37, 42, 38, 43, 39, 44, 40, 45, 51, 46, 52, 47, 53, 48, 54, 49, 55, 50, 61, 56, 62, 57, 63, 58, 64, 59, 65, 60, 66, 73, 67, 74

Offset: 1

Clark Kimberling, Sep 05 2011

See A194832 for a general discussion.

			Northwest corner:
1...3...5...9...13..19
2...4...7...11..16..22
6...10..14..20..26..34
8...12..17..23..30..38
15..21..27..35..43..53

Cf. A194832, A194902, A194903.

r = -Sqrt[12];
t[n_] := Table[FractionalPart[k*r], {k, 1, n}];
f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@
Sort[t[n], Less]], {n, 1,
    20}]] (* A194902 *)
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}]] (* A194903 *)
q[n_] := Position[p, n]; Flatten[Table[q[n],
{n, 1, 80}]]  (* A194904 *)

A194906 Rectangular array, by antidiagonals: row n gives the positions of n in the fractal sequence A194905; an interspersion.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 29, 38, 40, 41, 42, 43, 44, 45, 37, 39, 47, 49, 51, 52, 53, 54, 55, 46, 48, 50, 57, 59, 61, 63, 64, 65, 66, 56, 58, 60, 62, 68, 70, 72

Offset: 1

Clark Kimberling, Sep 05 2011

See A194832 for a general discussion.

			Northwest corner:
1...2...4...7...11..16..22
3...5...8...12..17..23..31
6...9...13..18..24..32..41
10..14..19..25..33..42..52
15..20..26..34..43..53..64

Cf. A194832, A194905, 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 *)

A194908 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, 2, 1, 3, 2, 1, 4, 3, 2, 1, 5, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 7, 6, 5, 4, 3, 2, 1, 7, 6, 5, 4, 3, 2, 1, 8, 7, 6, 5, 4, 3, 2, 9, 1, 8, 7, 6, 5, 4, 3, 10, 2, 9, 1, 8, 7, 6, 5, 4, 11, 3, 10, 2, 9, 1, 8, 7, 6, 5, 12, 4, 11, 3, 10, 2, 9, 1, 8, 7, 6, 13, 5, 12, 4, 11, 3, 10, 2, 9, 1, 8, 7, 14, 6

Offset: 1

Clark Kimberling, Sep 05 2011

See A194832 for a general discussion.

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

Cf. A194832, A194909, A194910.

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}]]  (* A194908 *)
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}]]  (* A194909 *)
q[n_] := Position[p, n]; Flatten[Table[q[n],
{n, 1, 80}]]  (* A194910 *)

A194909 Rectangular array, by antidiagonals: row n gives the positions of n in the fractal sequence A194905; an interspersion.

Original entry on oeis.org

1, 3, 2, 6, 5, 4, 10, 9, 8, 7, 15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 28, 27, 26, 25, 24, 23, 22, 35, 34, 33, 32, 31, 30, 29, 36, 44, 42, 41, 40, 39, 38, 37, 45, 43, 54, 52, 50, 49, 48, 47, 46, 55, 53, 51, 65, 63, 61, 59, 58, 57, 56, 66, 64, 62, 60, 77, 75, 73

Offset: 1

Clark Kimberling, Sep 05 2011

See A194832 for a general discussion.

			Northwest corner:
1...3...6...10..15..21
2...5...9...14..20..27
4...8...13..19..26..33
7...12..18..25..32..40
11..17..24..31..39..48
16..23..30..38..47..57

Cf. A194832, A194908, A194910.

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}]]  (* A194908 *)
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}]]  (* A194909 *)
q[n_] := Position[p, n]; Flatten[Table[q[n],
{n, 1, 80}]]  (* A194910 *)

A194911 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=2^(1/3).

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 05 2011

See A194832 for a general discussion. The triangle is not equal to A194841.

			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, A194912, A194913.

r = 2^(1/3);
t[n_] := Table[FractionalPart[k*r], {k, 1, n}];
f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@
Sort[t[n], Less]], {n, 1, 20}]]  (* A194911 *)
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}]]  (* A194912 *)
q[n_] := Position[p, n]; Flatten[Table[q[n],
{n, 1, 80}]]  (* A194913 *)

A194912 Rectangular array, by antidiagonals: row n gives the positions of n in the fractal sequence A194905; 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 05 2011

See A194832 for a general discussion. A194912 is not equal to A194842.

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

Cf. A194832, A194911, A194913.

r = 2^(1/3);
t[n_] := Table[FractionalPart[k*r], {k, 1, n}];
f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@
Sort[t[n], Less]], {n, 1, 20}]]  (* A194911 *)
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}]]  (* A194912 *)
q[n_] := Position[p, n]; Flatten[Table[q[n],
{n, 1, 80}]]  (* A194913 *)

cp's OEIS Frontend

A194897 Rectangular array, by antidiagonals: row n gives the positions of n in the fractal sequence A194896; an interspersion.

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A194899 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(12).

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PARI

A194900 Rectangular array, by antidiagonals: row n gives the positions of n in the fractal sequence A194899; an interspersion.

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A194902 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(12).

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A194903 Rectangular array, by antidiagonals: row n gives the positions of n in the fractal sequence A194902; an interspersion.

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A194906 Rectangular array, by antidiagonals: row n gives the positions of n in the fractal sequence A194905; an interspersion.

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A194908 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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A194909 Rectangular array, by antidiagonals: row n gives the positions of n in the fractal sequence A194905; an interspersion.

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A194911 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=2^(1/3).

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