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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A194976 Fractalization of (1+[n/sqrt(2)]), 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, 9, 10, 6, 3, 1, 2, 4, 5, 7, 8, 9, 11, 10, 6, 3, 1, 2, 4, 5, 7, 8, 9, 11, 12, 10, 6, 3, 1, 2, 4, 5, 7, 8, 9, 11, 12, 13, 10, 6, 3, 1, 2, 4
Offset: 1

Views

Author

Clark Kimberling, Sep 07 2011

Keywords

Comments

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. The sequence (1+[n/sqrt(2)]) is A049474.

Links

Crossrefs

Cf. A194959, A049474, A194977, A194978.

Programs

  • Mathematica
    r = Sqrt[2]; p[n_] := 1 + Floor[n/r]
Table[p[n], {n, 1, 90}]  (* A049474 *)
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] (* A194976 *)
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}]] (* A194977 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]] (* A194978 *)

A195071 Inverse permutation of A194922; every positive integer occurs exactly once.

Original entry on oeis.org

1, 3, 2, 6, 5, 4, 9, 10, 8, 7, 13, 15, 14, 12, 11, 18, 21, 20, 19, 17, 16, 24, 27, 28, 26, 25, 23, 22, 31, 34, 36, 35, 33, 32, 30, 29, 39, 42, 45, 44, 43, 41, 40, 38, 37, 48, 51, 55, 54, 53, 52, 50, 49, 47, 46, 58, 61, 65, 66, 64, 63, 62, 60, 59, 57, 56, 69, 72, 76
Offset: 1

Views

Author

Clark Kimberling, Sep 08 2011

Keywords

Links

Crossrefs

Cf. A194922, A194921, A194920, A194959.

Programs

  • Mathematica
    r = Sqrt[2]; p[n_] := n - Floor[n/r]
Table[p[n], {n, 1, 90}]  (* A194920 *)
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]  (* A194921 *)
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}]]  (* A194922 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]]   (* A195071 *)

A195073 Fractalization of (n-[n/sqrt(3)]), where [ ]=floor.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 2, 4, 3, 1, 2, 4, 5, 3, 1, 2, 4, 6, 5, 3, 1, 2, 4, 7, 6, 5, 3, 1, 2, 4, 7, 8, 6, 5, 3, 1, 2, 4, 7, 9, 8, 6, 5, 3, 1, 2, 4, 7, 9, 10, 8, 6, 5, 3, 1, 2, 4, 7, 9, 11, 10, 8, 6, 5, 3, 1, 2, 4, 7, 9, 11, 12, 10, 8, 6, 5, 3, 1, 2, 4, 7, 9, 11, 13, 12, 10, 8, 6, 5, 3, 1, 2, 4, 7, 9
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 (n-[n/sqrt(3)]) is A195072.

Crossrefs

Cf. A195072, A194974, A195075.

Programs

  • Mathematica
    r = Sqrt[3]; p[n_] := n - Floor[n/r]
Table[p[n], {n, 1, 90}]   (* A195072 *)
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]   (* A195073 *)
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}]]    (* A195074 *)
q[n_] := Position[w, n]; Flatten[
Table[q[n], {n, 1, 80}]]   (* A195075 *)

A195074 Interspersion fractally induced by A194920, a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 3, 2, 6, 4, 5, 10, 7, 9, 8, 15, 11, 14, 12, 13, 21, 16, 20, 17, 19, 18, 28, 22, 27, 23, 26, 25, 24, 36, 29, 35, 30, 34, 33, 31, 32, 45, 37, 44, 38, 43, 42, 39, 41, 40, 55, 46, 54, 47, 53, 52, 48, 51, 49, 50, 66, 56, 65, 57, 64, 63, 58, 62, 59, 61, 60, 78, 67, 77
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, A194974 is a permutation of the positive integers, with inverse A195075.
To see that A195074 differs from A194988, note that the generating sequences A195072 and A194986 differ.

Examples

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

Crossrefs

Cf. A195072, A195074, A195075.

Programs

  • Mathematica
    r = Sqrt[3]; p[n_] := n - Floor[n/r]
Table[p[n], {n, 1, 90}]   (* A195072 *)
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]   (* A195073 *)
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}]]    (* A195074 *)
q[n_] := Position[w, n]; Flatten[
Table[q[n], {n, 1, 80}]]   (* A195075 *)

A195107 Fractalization of the fractal sequence A004736. Interspersion fractally induced by A004736.

Original entry on oeis.org

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

Crossrefs

Cf. A194959, A004736, A195108, 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 *)

A195110 Fractalization of the fractal sequence A002260. Interspersion fractally induced by A002260.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 4, 2, 3, 1, 4, 5, 2, 3, 1, 4, 5, 6, 2, 3, 1, 7, 4, 5, 6, 2, 3, 1, 7, 8, 4, 5, 6, 2, 3, 1, 7, 8, 9, 4, 5, 6, 2, 3, 1, 7, 8, 9, 10, 4, 5, 6, 2, 3, 1, 11, 7, 8, 9, 10, 4, 5, 6, 2, 3, 1, 11, 12, 7, 8, 9, 10, 4, 5, 6, 2, 3, 1, 11, 12, 13, 7, 8, 9, 10, 4, 5, 6, 2, 3, 1, 11, 12
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 A002260 is the fractal sequence obtained by concatenating the segments 1; 12; 123; 1234; 12345;...

Crossrefs

Cf. A194959, A002260, A195111, A195112.

Programs

  • Mathematica
    j[n_] := Table[k, {k, 1, n}]; t[1] = j[1];
t[n_] := Join[t[n - 1], j[n]]   (* A002260 *)
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] (* A195110 *)
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}]] (* A195111 *)
q[n_] := Position[w, n]; Flatten[Table[q[n], {n, 1, 80}]]  (* A195112 *)

A195111 Interspersion fractally induced by the fractal sequence A002260.

Original entry on oeis.org

1, 3, 2, 6, 4, 5, 10, 8, 9, 7, 15, 13, 14, 11, 12, 21, 19, 20, 16, 17, 18, 28, 26, 27, 23, 24, 25, 22, 36, 34, 35, 31, 32, 33, 29, 30, 45, 43, 44, 40, 41, 42, 37, 38, 39, 55, 53, 54, 50, 51, 52, 46, 47, 48, 49, 66, 64, 65, 61, 62, 63, 57, 58, 59, 60, 56, 78, 76, 77
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, A194111 is a permutation of the positive integers, with inverse A195129.
The sequence A002260 is the fractal sequence obtained by concatenating the segments 1; 12; 123; 1234; 12345;...

Examples

			Northwest corner:
1...3...6...10..15..21..28..36..45
2...4...8...13..19..26..34..43..53
5...9...14..20..27..35..44..54..65
7...11..16..23..31..40..50..61..73
12..17..24..32..41..51..62..74..87

Crossrefs

Cf. A194959, A002260, A195110, A195112.

Programs

  • Mathematica
    j[n_] := Table[k, {k, 1, n}]; t[1] = j[1];
t[n_] := Join[t[n - 1], j[n]]   (* A002260 *)
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] (* A195110 *)
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}]] (* A195111 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
  {n, 1, 80}]]  (* A195112 *)

A194916 Inverse permutation of A194915; every positive integer occurs exactly once.

Original entry on oeis.org

1, 3, 2, 5, 6, 4, 8, 10, 9, 7, 12, 15, 14, 13, 11, 17, 20, 21, 19, 18, 16, 23, 26, 28, 27, 25, 24, 22, 30, 33, 36, 35, 34, 32, 31, 29, 38, 41, 44, 45, 43, 42, 40, 39, 37, 47, 50, 53, 55, 54, 52, 51, 49, 48, 46, 57, 60, 63, 66, 65, 64, 62, 61, 59, 58, 56, 68, 71, 74
Offset: 1

Views

Author

Clark Kimberling, Sep 08 2011

Keywords

Crossrefs

Cf. A194915, A194914, A194959.

Programs

A194917 Fractalization of (n-[nr-n]), where [ ]=floor and r=(1+sqrt(5))/2 (the golden ratio).

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 2, 4, 3, 1, 2, 5, 4, 3, 1, 2, 5, 6, 4, 3, 1, 2, 5, 7, 6, 4, 3, 1, 2, 5, 7, 8, 6, 4, 3, 1, 2, 5, 7, 9, 8, 6, 4, 3, 1, 2, 5, 7, 10, 9, 8, 6, 4, 3, 1, 2, 5, 7, 10, 11, 9, 8, 6, 4, 3, 1, 2, 5, 7, 10, 12, 11, 9, 8, 6, 4, 3, 1, 2, 5, 7, 10, 13, 12, 11, 9, 8, 6, 4, 3, 1, 2, 5, 7
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 (n-[nr-n]) is A189663.

Crossrefs

Cf. A194959, A189663, A194918, A194919.

Programs

  • Mathematica
    r = GoldenRatio; p[n_] := n - Floor[n/r]
Table[p[n], {n, 1, 90}]  (* A189663 *)
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] (*  A194917 *)
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}]] (* A194918 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]] (* A194919 *)

A194918 Interspersion fractally induced by A189663, a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 3, 2, 6, 4, 5, 10, 7, 9, 8, 15, 11, 14, 13, 12, 21, 16, 20, 19, 17, 18, 28, 22, 27, 26, 23, 25, 24, 36, 29, 35, 34, 30, 33, 31, 32, 45, 37, 44, 43, 38, 42, 39, 41, 40, 55, 46, 54, 53, 47, 52, 48, 51, 50, 49, 66, 56, 65, 64, 57, 63, 58, 62, 61, 59, 60, 78, 67, 77
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, A194918 is a permutation of the positive integers, with inverse A194919.

Examples

			Northwest corner:
1...3...6...10..15..21
2...4...7...11..16..22
5...9...14..20..27..35
8...13..19..26..34..43
12..17..23..30..38..47

Crossrefs

Cf. A194959, A189663, A194917, A194919.

Programs

  • Mathematica
    r = GoldenRatio; p[n_] := n - Floor[n/r]
Table[p[n], {n, 1, 90}]  (* A189663 *)
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] (*  A194917 *)
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}]] (* A194918 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]] (* A194919 *)
Previous Showing 21-30 of 61 results. Next

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