cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A195080 Interspersion fractally induced by A008621, a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 3, 2, 6, 5, 4, 10, 9, 7, 8, 15, 14, 11, 13, 12, 21, 20, 16, 19, 18, 17, 28, 27, 22, 26, 25, 24, 23, 36, 35, 29, 34, 33, 32, 30, 31, 45, 44, 37, 43, 42, 41, 38, 40, 39, 55, 54, 46, 53, 52, 51, 47, 50, 49, 48, 66, 65, 56, 64, 63, 62, 57, 61, 60, 59, 58, 78, 77, 67
Offset: 1

Views

Author

Clark Kimberling, Sep 08 2011

Keywords

Comments

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. Every pair of rows eventually intersperse. As a sequence, A194980 is a permutation of the positive integers, with inverse A195081.

Examples

			Northwest corner:
1...3...6...10..15..21..38
2...5...9...14..20..27..35
4...7...11..16..22..29..37
8...13..19..26..34..43..53
12..18..25..33..42..52..63

Crossrefs

Cf. A008621, A195079, A195081.

Programs

  • Mathematica
    r = 4; p[n_] := 1 + Floor[n/r]
Table[p[n], {n, 1, 90}]  (* A008621 *)
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20] (* A195079 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13},
{k, 1, n}]] (* A195080 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]] (* A195081 *)

A195082 Fractalization of (1+[2*n/3]), where [ ] = floor.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 08 2011

Keywords

Comments

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. The sequence (1+[2*n/3]) is essentially A004396.

Crossrefs

Cf. A004396, A195083, A195096.

Programs

  • Mathematica
    r = 2/3; p[n_] := 1 + Floor[n*r]
Table[p[n], {n, 1, 90}]  (* ess A004396 *)
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20] (* A195082 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13}, {k, 1, n}]] (* A195083 *)
q[n_] := Position[w, n]; Flatten[Table[q[n], {n, 1, 80}]] (* A195096 *)

A195083 Interspersion fractally induced by (1+[2*n/3]), where [ ] = floor; a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 9, 11, 12, 15, 13, 14, 16, 17, 21, 18, 19, 20, 22, 23, 28, 24, 25, 27, 26, 29, 30, 36, 31, 32, 35, 33, 34, 37, 38, 45, 39, 40, 44, 41, 42, 43, 46, 47, 55, 48, 49, 54, 50, 51, 53, 52, 56, 57, 66, 58, 59, 65, 60, 61, 64, 62, 63, 67, 68, 78
Offset: 1

Views

Author

Clark Kimberling, Sep 08 2011

Keywords

Comments

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. Every pair of rows eventually intersperse. As a sequence, A194983 is a permutation of the positive integers, with inverse A195096.

Examples

			Northwest corner:
  1   2   4   7   11  16
  3   5   8   12  17  23
  6   10  15  21  28  36
  9   13  18  24  31  39
  14  19  25  32  40  49

Crossrefs

Cf. A004396, A195082, A195096.

Programs

  • Mathematica
    r = 2/3; p[n_] := 1 + Floor[n*r]
Table[p[n], {n, 1, 90}]  (* ess A004396 *)
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20] (* A195082 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13}, {k, 1, n}]] (* A195083 *)
q[n_] := Position[w, n]; Flatten[Table[q[n], {n, 1, 80}]] (* A195096 *)

A195097 Fractalization of (1+[3n/4]), where [ ] = floor.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 08 2011

Keywords

Comments

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. The sequence (1+[3n/4]) is a subsequence ofy A037915.

Crossrefs

Cf. A194959, A002265, A195098, A195099.

Programs

  • Mathematica
    r = 3/4; p[n_] := 1 + Floor[n*r] (* A037915 *)
Table[p[n], {n, 1, 90}]
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20]    (* A195097 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13},
{k, 1, n}]](* A195098 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]](* A195099 *)

A195098 Interspersion fractally induced by (1+[3n/4]), where [ ] = floor; a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 14, 16, 17, 18, 21, 19, 20, 22, 23, 24, 28, 25, 26, 27, 29, 30, 31, 36, 32, 33, 34, 35, 37, 38, 39, 45, 40, 41, 42, 44, 43, 46, 47, 48, 55, 49, 50, 51, 54, 52, 53, 56, 57, 58, 66, 59, 60, 61, 65, 62, 63, 64, 67, 68, 69
Offset: 1

Views

Author

Clark Kimberling, Sep 08 2011

Keywords

Comments

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. Every pair of rows eventually intersperse. As a sequence, A194998 is a permutation of the positive integers, with inverse A195099.

Examples

			Northwest corner:
1...2...4...7...11..16..22
3...5...8...12..17..23..30
6...9...13..18..24..31..39
10..15..21..28..36..45..55
14..19..25..32..40..49..59

Crossrefs

Cf. A194959, A037915, A195097, A195099.

Programs

  • Mathematica
    r = 3/4; p[n_] := 1 + Floor[n*r] (* A037915 *)
Table[p[n], {n, 1, 90}]
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20]    (* A195097 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13},
{k, 1, n}]](* A195098 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]](* A195099 *)

A195108 Interspersion fractally induced by A004736.

Original entry on oeis.org

1, 2, 3, 5, 6, 4, 8, 10, 7, 9, 13, 15, 11, 14, 12, 19, 21, 17, 20, 18, 16, 26, 28, 23, 27, 24, 22, 25, 34, 36, 30, 35, 32, 29, 33, 31, 43, 45, 39, 44, 41, 37, 42, 40, 38, 53, 55, 49, 54, 51, 47, 52, 50, 48, 46, 64, 66, 59, 65, 62, 57, 63, 61, 58, 56, 60, 76, 78, 71
Offset: 1

Views

Author

Clark Kimberling, Sep 09 2011

Keywords

Comments

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence.
The sequence A004736 is the fractal sequence obtained by concatenating the segments 1; 2,1; 3,2,1; 4,3,2,1;...
Every pair of rows of A195108 eventually intersperse.
As a sequence, A194108 is a permutation of the positive integers, with inverse A195109.

Examples

			Northwest corner:
1...2...5...8...13..19..26
3...6...10..15..21..28..36
4...7...11..17..23..30..39
9...14..20..27..35..44..54
12..18..24..32..41..51..62

Crossrefs

Cf. A194959, A004736, A195107, A195109.

Programs

  • Mathematica
    j[n_] := Table[n + 1 - k, {k, 1, n}]; t[1] = j[1];
t[n_] := Join[t[n - 1], j[n]]   (* A004736 *)
t[10]
p[n_] := t[20][[n]]
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20] (* A195107 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13},
{k, 1, n}]] (* A195108 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]] (* A195109 *)

A195113 Fractalization of the fractal sequence obtained by deleting the second two terms of the fractal sequence A002260.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 09 2011

Keywords

Comments

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence p; for the present case, p is the concatenation of the segments 1, 123,1234,12345,123456,..., so that p is obtained from A002260 by deleting the segment 12.

Crossrefs

Cf. A194959, A002260, A195114, A195115.

Programs

  • Mathematica
    j[n_] := Table[k, {k, 1, n}];
t[1] = j[1]; t[2] = j[1];
t[n_] := Join[t[n - 1], j[n]] (* A002260; initial 1,1,2 repl by 1 *)
t[12]
p[n_] := t[20][[n]]
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20]  (* A195113 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13},
{k, 1, n}]] (* A195114 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]]  (* A195115 *)

A195114 Interspersion fractally induced by the fractal sequence obtained by deleting the second two terms of the fractal sequence A002260.

Original entry on oeis.org

1, 3, 2, 6, 4, 5, 10, 7, 8, 9, 15, 12, 13, 14, 11, 21, 18, 19, 20, 16, 17, 28, 25, 26, 27, 22, 23, 24, 36, 33, 34, 35, 29, 30, 31, 32, 45, 42, 43, 44, 38, 39, 40, 41, 37, 55, 52, 53, 54, 48, 49, 50, 51, 46, 47, 66, 63, 64, 65, 59, 60, 61, 62, 56, 57, 58, 78, 75, 76
Offset: 1

Views

Author

Clark Kimberling, Sep 09 2011

Keywords

Comments

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. Every pair of rows eventually intersperse. As a sequence, A194114 is a permutation of the positive integers, with inverse A195115.

Examples

			Northwest corner:
1...3...6...10..15..21..28
2...4...7...12..18..25..33
5...8...13..19..26..34..43
9...14..20..27..35..44..54
11..16..22..29..38..48..59

Crossrefs

Cf. A194959, A002260, A195113, A195115, A195111.

Programs

  • Mathematica
    j[n_] := Table[k, {k, 1, n}];
t[1] = j[1]; t[2] = j[1];
t[n_] := Join[t[n - 1], j[n]] (* A002260; initial 1,1,2 repl by 1 *)
t[12]
p[n_] := t[20][[n]]
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20]  (* A195113 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13},
{k, 1, n}]] (* A195114 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]]  (* A195115 *)

A195183 Fractalization of the prime marker sequence A089026.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 10 2011

Keywords

Comments

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. (The prime marker sequence A089026 is defined by p(n)=n if n is prime and p(n)=1 otherwise.)

Links

Crossrefs

Cf. A194959, A089026, A195184, A195185.

Programs

  • Mathematica
    Table[p[n], {n, 1, 90}]  (* A089026 *)
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20] (* A195183 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13}, {k, 1, n}]]  (* A195184 *)
q[n_] := Position[w, n]; Flatten[Table[q[n], {n, 1, 80}]]  (* A195185 *)

A195184 Interspersion fractally induced by the prime marker sequence A089026.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 7, 12, 13, 14, 11, 15, 18, 19, 20, 17, 21, 16, 24, 25, 26, 23, 27, 22, 28, 32, 33, 34, 31, 35, 30, 36, 29, 41, 42, 43, 40, 44, 39, 45, 38, 37, 51, 52, 53, 50, 54, 49, 55, 48, 47, 46, 61, 62, 63, 60, 64, 59, 65, 58, 57, 56, 66, 73, 74, 75
Offset: 1

Views

Author

Clark Kimberling, Sep 10 2011

Keywords

Comments

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. Every pair of rows eventually intersperse. As a sequence, A194184 is a permutation of the positive integers, with inverse A195185. (The prime marker sequence A089026 is given by p(n)=n if n is prime and p(n)=1 otherwise).

Examples

			Northwest corner:
1...2...4...8...12..18..24..32
3...5...9...13..19..26..33..42
6...10..14..20..26..34..43..53
7...11..17..23..31..40..50..60
15..21..27..35..44..54..64..76

Links

Crossrefs

Cf. A194959, A089026, A195184, A195185.

Programs

  • Mathematica
    p[n_] := If[PrimeQ[n], n, 1]
Table[p[n], {n, 1, 90}]  (* A089026 *)
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20] (* A195183 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13}, {k, 1, n}]]  (* A195184 *)
q[n_] := Position[w, n]; Flatten[Table[q[n], {n, 1, 80}]]  (* A195185 *)
Previous Showing 41-50 of 61 results. Next

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