A194832 -id:A194832 - cp's OEIS frontend

A194838 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).

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

Offset: 1

Clark Kimberling, Sep 03 2011

See A194832 for a general discussion.

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

Cf. A194832, A194839, A194840.

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}]]  (* A194838 *)
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}]]  (* A194839 *)
q[n_] := Position[p, n]; Flatten[
Table[q[n], {n, 1, 80}]]  (* A194840 *)

A194839 Rectangular array, by antidiagonals: row n gives the positions of n in the fractal sequence A194838; an interspersion.

Original entry on oeis.org

1, 3, 2, 6, 5, 4, 9, 8, 7, 10, 14, 12, 11, 15, 13, 20, 18, 16, 21, 19, 17, 27, 25, 23, 28, 26, 24, 22, 34, 32, 30, 36, 33, 31, 29, 35, 43, 40, 38, 45, 42, 39, 37, 44, 41, 53, 50, 47, 55, 52, 49, 46, 54, 51, 48, 64, 61, 58, 66, 63, 60, 57, 65, 62, 59, 56, 75, 72, 69

Offset: 1

Clark Kimberling, Sep 03 2011

See A194832 and A194838.

			Northwest corner:
1...3...6...9...14..20..27
2...5...8...12..18..25..32
4...7...11..16..23..30..38
10..15..21..28..36..45..55
13..19..26..33..42..52..63
17..24..31..39..49..60..71

Cf. A194838, A194840, 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}]]  (* A194838 *)
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}]]  (* A194839 *)
q[n_] := Position[p, n]; Flatten[
 Table[q[n], {n, 1, 80}]]  (* A194840 *)

A194844 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(5).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 04 2011

See A194832 for a general discussion.

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

Cf. A194832, A194845, A194846.

r = Sqrt[5];
t[n_] := Table[FractionalPart[k*r], {k, 1, n}];
f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@ Sort[t[n], Less]], {n, 1, 20}]]  (* A194844 *)
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}]]    (* A194845 *)
q[n_] := Position[p, n]; Flatten[
 Table[q[n], {n, 1, 80}]]   (* A194846 *)

A194845 Rectangular array, by antidiagonals: row n gives the positions of n in the fractal sequence A194844; an interspersion.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 11, 17, 19, 20, 21, 16, 18, 23, 25, 27, 28, 22, 24, 26, 30, 32, 34, 36, 29, 31, 33, 35, 39, 41, 43, 45, 38, 40, 42, 44, 37, 48, 51, 53, 55, 47, 50, 52, 54, 46, 49, 58, 61, 64, 66, 57, 60, 63, 65, 56, 59, 62, 69, 72, 75

Offset: 1

Clark Kimberling, Sep 04 2011

See A194832 and A194844.

			Northwest corner:
1...2...4...7...12..17
3...5...8...13..19..25
6...9...14..20..27..34
10..15..21..28..36..45
11..16..22..29..38..47
18..24..31..40..50..60

Cf. A194832, A194844, A194846.

r = Sqrt[5];
t[n_] := Table[FractionalPart[k*r], {k, 1, n}];
f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@ Sort[t[n], Less]], {n, 1, 20}]]  (* A194844 *)
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}]]    (* A194845 *)
q[n_] := Position[p, n]; Flatten[
 Table[q[n], {n, 1, 80}]]   (* A194846 *)

A194856 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(5).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 04 2011

See A194832 for a general discussion.

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

Cf. A194832, A194857, A194858.

r = -Sqrt[5];
t[n_] := Table[FractionalPart[k*r], {k, 1, n}];
f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@
Sort[t[n], Less]], {n, 1, 20}]]  (* A194856 *)
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}]]  (* A194857 *)
q[n_] := Position[p, n]; Flatten[Table[q[n],
{n, 1, 80}]]  (* A194858 *)

A194857 Rectangular array, by antidiagonals: row n gives the positions of n in the fractal sequence A194856; an interspersion.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 04 2011

See A194832 for a general discussion.

			Northwest corner:
1...3...6...10...14...20
2...5...9...13...18...25
4...8...12..17...23...51
7...11..16..22...29...37
15..21..38..36...44...54

Cf. A194856, A194858, A194832.

r = -Sqrt[5];
t[n_] := Table[FractionalPart[k*r], {k, 1, n}];
f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@
Sort[t[n], Less]], {n, 1, 20}]]  (* A194856 *)
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}]]  (* A194857 *)
q[n_] := Position[p, n]; Flatten[Table[q[n],
{n, 1, 80}]]  (* A194858 *)

A194859 Triangular array (and fractal sequence): row n is the permutation of (1,2,...,n) obtained from the increasing ordering of fractional parts {e}, {2e}, ..., {ne}.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 04 2011

See A194832 for a general discussion.

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

Cf. A194832, A194860, A194861.

r = E;
t[n_] := Table[FractionalPart[k*r], {k, 1, n}];
f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@
Sort[t[n], Less]], {n, 1, 20}]]  (* A194859 *)
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}]]  (* A194860 *)
q[n_] := Position[p, n]; Flatten[Table[q[n],
{n, 1, 80}]]  (* A194861 *)

A194860 Rectangular array, by antidiagonals: row n gives the positions of n in the fractal sequence A194859; an interspersion.

Original entry on oeis.org

1, 3, 2, 6, 5, 4, 9, 8, 7, 10, 14, 12, 11, 15, 13, 20, 18, 16, 21, 19, 17, 27, 25, 23, 28, 26, 24, 22, 34, 32, 30, 36, 33, 31, 29, 35, 43, 40, 38, 45, 42, 39, 37, 44, 41, 53, 50, 47, 55, 52, 49, 46, 54, 51, 48, 63, 60, 57, 65, 62, 59, 56, 64, 61, 58, 66, 75, 71, 68

Offset: 1

Clark Kimberling, Sep 04 2011

See A194832 for a general discussion.

			Northwest corner:
1...3...6...9...14..20
2...5...8...12..18..25
4...7...11..16..23..30
10..15..21..28..36..45
13..19..26..33..42..52
17..24..31..39..49..59

Cf. A194859, A194861, A194832.

r = E;
t[n_] := Table[FractionalPart[k*r], {k, 1, n}];
f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@
Sort[t[n], Less]], {n, 1, 20}]]  (* A194859 *)
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}]]  (* A194860 *)
q[n_] := Position[p, n]; Flatten[Table[q[n],
{n, 1, 80}]]  (* A194861 *)

A194862 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, 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, 3, 6, 1, 4, 7, 2, 5, 8, 3, 6, 9, 1, 4, 7, 2, 5, 8, 3, 6, 9, 1, 4, 7, 10, 2, 5, 8, 11, 3, 6, 9, 1, 4, 7, 10, 2, 5, 8, 11, 3, 6, 9, 1, 12, 4, 7, 10, 2, 5, 8, 11, 3, 6, 9, 1, 12, 4, 7, 10, 2, 13, 5, 8, 11, 3, 14

Offset: 1

Clark Kimberling, Sep 04 2011

See A194832 for a general discussion.

			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
3 6 1 4 7 2 5 8
3 6 9 1 4 7 2 5 8

Cf. A194832, A194863, A194867.

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}]]  (* A194862 *)
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}]]  (* A194863 *)
q[n_] := Position[p, n]; Flatten[Table[q[n],
{n, 1, 80}]]  (* A194867 *)

A194863 Rectangular array, by antidiagonals: row n gives the positions of n in the fractal sequence A194862; an interspersion.

Original entry on oeis.org

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, 31, 34, 29, 32, 35, 30, 33, 36, 40, 43, 37, 41, 44, 38, 42, 45, 39, 49, 53, 46, 50, 54, 47, 51, 55, 48, 52, 60, 64, 57, 61, 65, 58, 62, 66, 59, 63, 56, 71, 76, 68

Offset: 1

Clark Kimberling, Sep 04 2011

For a general discussion, see A194832.

			Northwest corner:
1...2...5...8...12..18
3...6...10..14..20..27
4...7...11..16..22..29
9...13..19..25..32..41
15..21..28..35..44..54

Cf. A194832, A194862, A194867.

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}]]  (* A194862 *)
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}]]  (* A194863 *)
q[n_] := Position[p, n]; Flatten[Table[q[n],
{n, 1, 80}]]  (* A194867 *)

cp's OEIS Frontend

A194838 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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A194839 Rectangular array, by antidiagonals: row n gives the positions of n in the fractal sequence A194838; an interspersion.

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A194844 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(5).

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A194845 Rectangular array, by antidiagonals: row n gives the positions of n in the fractal sequence A194844; an interspersion.

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A194856 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(5).

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A194857 Rectangular array, by antidiagonals: row n gives the positions of n in the fractal sequence A194856; an interspersion.

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A194859 Triangular array (and fractal sequence): row n is the permutation of (1,2,...,n) obtained from the increasing ordering of fractional parts {e}, {2e}, ..., {ne}.

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A194860 Rectangular array, by antidiagonals: row n gives the positions of n in the fractal sequence A194859; an interspersion.

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A194862 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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