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.

Showing 1-8 of 8 results.

A287557 Positions of 0 in A287556.

Original entry on oeis.org

1, 8, 10, 15, 20, 22, 27, 29, 35, 37, 44, 46, 50, 55, 57, 64, 68, 70, 75, 77, 83, 85, 92, 94, 98, 103, 105, 112, 113, 120, 122, 127, 130, 135, 137, 144, 145, 152, 154, 159, 164, 166, 171, 173, 179, 181, 188, 190, 195, 197, 204, 206, 210, 215, 217, 224, 225
Offset: 1

Views

Author

Clark Kimberling, May 31 2017

Keywords

Comments

a(n) - a(n-1) is in {1,2,3,4,5,6,7} for n >= 1; also, 4n - a(n) is in {0,1,2,3} for n >= 1. The first 20 numbers 4n - a(n) are 3, 0, 2, 1, 0, 2, 1, 3, 1, 3, 0, 2, 2, 1, 3, 0, 0, 2, 1, 3, with
0 in positions given by A287558,
1 in positions given by A287559,
2 in positions given by A287560,
3 in positions given by A287557.

Links

Crossrefs

Cf. A287556, A287558, A287559, A287560.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {0, 1, 3, 2}, 1 -> {1, 3, 2, 0}, 2 -> {3, 2, 0, 1}, 3 -> {2, 0, 1, 3}}] &, {0}, 9];   (* A287556 *)
Flatten[Position[s, 0]]; (* A287557 *)
Flatten[Position[s, 1]]; (* A287558 *)
Flatten[Position[s, 2]]; (* A287559 *)
Flatten[Position[s, 3]]; (* A287560 *)

A287558 Positions of 1 in A287556.

Original entry on oeis.org

2, 5, 11, 16, 17, 23, 28, 30, 36, 38, 41, 47, 51, 56, 58, 61, 65, 71, 76, 78, 84, 86, 89, 95, 99, 104, 106, 109, 114, 117, 123, 128, 131, 136, 138, 141, 146, 149, 155, 160, 161, 167, 172, 174, 180, 182, 185, 191, 196, 198, 201, 207, 211, 216, 218, 221, 226
Offset: 1

Views

Author

Clark Kimberling, May 31 2017

Keywords

Comments

a(n) - a(n-1) is in {1,2,3,4,5,6,7} for n >= 1; also, 4n - a(n) is in {0,1,2,3} for n >= 1. The first 20 numbers 4n - a(n) are 2, 3, 1, 0, 3, 1, 0, 2, 0, 2, 3, 1, 1, 0, 2, 3, 3, 1, 0, 2, with
0 in positions given by A287559,
1 in positions given by A287560,
2 in positions given by A287557,
3 in positions given by A287558.

Links

Crossrefs

Cf. A287556, A287557, A287559, A287560.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {0, 1, 3, 2}, 1 -> {1, 3, 2, 0}, 2 -> {3, 2, 0, 1}, 3 -> {2, 0, 1, 3}}] &, {0}, 9];   (* A287556 *)
Flatten[Position[s, 0]]; (* A287557 *)
Flatten[Position[s, 1]]; (* A287558 *)
Flatten[Position[s, 2]]; (* A287559 *)
Flatten[Position[s, 3]]; (* A287560 *)

A287559 Positions of 2 in A287556.

Original entry on oeis.org

4, 7, 9, 14, 19, 21, 26, 32, 34, 40, 43, 45, 49, 54, 60, 63, 67, 69, 74, 80, 82, 88, 91, 93, 97, 102, 108, 111, 116, 119, 121, 126, 129, 134, 140, 143, 148, 151, 153, 158, 163, 165, 170, 176, 178, 184, 187, 189, 194, 200, 203, 205, 209, 214, 220, 223, 228
Offset: 1

Views

Author

Clark Kimberling, May 31 2017

Keywords

Comments

a(n) - a(n-1) is in {1,2,3,4,5,6,7} for n >= 1; also, 4n - a(n) is in {0,1,2,3} for n >= 1. The first 20 numbers 4n - a(n) are 0, 1, 3, 2, 1, 3, 2, 0, 2, 0, 1, 3, 3, 2, 0, 1, 1, 3, 2, 0, with
0 in positions given by A287557,
1 in positions given by A287558,
2 in positions given by A287560,
3 in positions given by A287559.

Links

Crossrefs

Cf. A287556, A287557, A287558, A287560.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {0, 1, 3, 2}, 1 -> {1, 3, 2, 0}, 2 -> {3, 2, 0, 1}, 3 -> {2, 0, 1, 3}}] &, {0}, 9];   (* A287556 *)
Flatten[Position[s, 0]]; (* A287557 *)
Flatten[Position[s, 1]]; (* A287558 *)
Flatten[Position[s, 2]]; (* A287559 *)
Flatten[Position[s, 3]]; (* A287560 *)

A287560 Positions of 3 in A287556.

Original entry on oeis.org

3, 6, 12, 13, 18, 24, 25, 31, 33, 39, 42, 48, 52, 53, 59, 62, 66, 72, 73, 79, 81, 87, 90, 96, 100, 101, 107, 110, 115, 118, 124, 125, 132, 133, 139, 142, 147, 150, 156, 157, 162, 168, 169, 175, 177, 183, 186, 192, 193, 199, 202, 208, 212, 213, 219, 222, 227
Offset: 1

Views

Author

Clark Kimberling, May 31 2017

Keywords

Comments

a(n) - a(n-1) is in {1,2,3,4,5,6,7} for n >= 1; also, 4n - a(n) is in {0,1,2,3} for n >= 1. The first 20 numbers 4n - a(n) are 1, 2, 0, 3, 2, 0, 3, 1, 3, 1, 2, 0, 0, 3, 1, 2, 2, 0, 3, 1, with
0 in positions given by A287560,
1 in positions given by A287557,
2 in positions given by A287558,
3 in positions given by A287559.

Links

Crossrefs

Cf. A287556, A287557, A287558, A287559 .

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {0, 1, 3, 2}, 1 -> {1, 3, 2, 0}, 2 -> {3, 2, 0, 1}, 3 -> {2, 0, 1, 3}}] &, {0}, 9];   (* A287556 *)
Flatten[Position[s, 0]]; (* A287557 *)
Flatten[Position[s, 1]]; (* A287558 *)
Flatten[Position[s, 2]]; (* A287559 *)
Flatten[Position[s, 3]]; (* A287560 *)

A287561 Start with 0 and repeatedly substitute 0->0213, 1->2130, 2->1302, 3->3021.

Original entry on oeis.org

0, 2, 1, 3, 1, 3, 0, 2, 2, 1, 3, 0, 3, 0, 2, 1, 2, 1, 3, 0, 3, 0, 2, 1, 0, 2, 1, 3, 1, 3, 0, 2, 1, 3, 0, 2, 2, 1, 3, 0, 3, 0, 2, 1, 0, 2, 1, 3, 3, 0, 2, 1, 0, 2, 1, 3, 1, 3, 0, 2, 2, 1, 3, 0, 1, 3, 0, 2, 2, 1, 3, 0, 3, 0, 2, 1, 0, 2, 1, 3, 3, 0, 2, 1, 0, 2
Offset: 1

Views

Author

Clark Kimberling, May 31 2017

Keywords

Comments

This is the fixed point of the morphism 0->0213, 1->2130, 2->1302, 3->3021 starting with 0. Let t be the (nonperiodic) sequence of positions of 0, and likewise, u for 1, v for 2, and w for 3; then t(n)/n -> 4, u(n)/n -> 4, v(n)/n -> 4, w(n)/n -> 4. Also, t(n) + u(n) + v(n) + w(n) = 16*n - 6 for n >= 1. See A287556 for a guide to related sequences.

Examples

			First three iterations of the morphism:
0213
0213130221303021
0213130221303021213030210213130213022130302102133021021313022130

Links

Crossrefs

Cf. A287562, A287563, A287564, A287565.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {0, 2, 1, 3}, 1 -> {2, 1, 3, 0}, 2 -> {1, 3, 0, 2}, 3 -> {3, 0, 2, 1}}] &, {0}, 9];   (* A287561 *)
Flatten[Position[s, 0]]; (* A287562 *)
Flatten[Position[s, 1]]; (* A287563 *)
Flatten[Position[s, 2]]; (* A287564 *)
Flatten[Position[s, 3]]; (* A287565 *)

Formula

a(n) = 4n - A287565(n) for n >= 1.

A287566 Start with 0 and repeatedly substitute 0->0231, 1->2310, 2->3102, 3->1023.

Original entry on oeis.org

0, 2, 3, 1, 3, 1, 0, 2, 1, 0, 2, 3, 2, 3, 1, 0, 1, 0, 2, 3, 2, 3, 1, 0, 0, 2, 3, 1, 3, 1, 0, 2, 2, 3, 1, 0, 0, 2, 3, 1, 3, 1, 0, 2, 1, 0, 2, 3, 3, 1, 0, 2, 1, 0, 2, 3, 2, 3, 1, 0, 0, 2, 3, 1, 2, 3, 1, 0, 0, 2, 3, 1, 3, 1, 0, 2, 1, 0, 2, 3, 3, 1, 0, 2, 1, 0
Offset: 1

Views

Author

Clark Kimberling, May 31 2017

Keywords

Comments

This is the fixed point of the morphism 0->0231, 1->2310, 2->3102, 3->1023 starting with 0. Let t be the (nonperiodic) sequence of positions of 0, and likewise, u for 1, v for 2, and w for 3; then t(n)/n -> 4, u(n)/n -> 4, v(n)/n -> 4, w(n)/n -> 4. Also, t(n) + u(n) + v(n) + w(n) = 16*n - 6 for n >= 1. See A287556 for a guide to related sequences.

Examples

			First three iterations of the morphism:
0231
0231310210232310
0231310210232310102323100231310223100231310210233102102323100231

Links

Crossrefs

Cf. A287567, A287568, A287569, A287570.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {0, 2, 3, 1}, 1 -> {2, 3, 2, 0}, 2 -> {3, 1, 0, 2}, 3 -> {1, 0, 2, 3}}] &, {0}, 9];   (* A287566 *)
Flatten[Position[s, 0]]; (* A287567 *)
Flatten[Position[s, 1]]; (* A287568 *)
Flatten[Position[s, 2]]; (* A287569 *)
Flatten[Position[s, 3]]; (* A287570 *)
Flatten[SubstitutionSystem[{0->{0,2,3,1},1->{2,3,1,0},2->{3,1,0,2},3->{1,0,2,3}},{0},{4}]] (* Harvey P. Dale, Jul 20 2023 *)

Formula

a(n) = 4n - A287568(n) for n >= 1.

A287571 Start with 0 and repeatedly substitute 0->0312, 1->3120, 2->1203, 3->2031.

Original entry on oeis.org

0, 3, 1, 2, 2, 0, 3, 1, 3, 1, 2, 0, 1, 2, 0, 3, 1, 2, 0, 3, 0, 3, 1, 2, 2, 0, 3, 1, 3, 1, 2, 0, 2, 0, 3, 1, 3, 1, 2, 0, 1, 2, 0, 3, 0, 3, 1, 2, 3, 1, 2, 0, 1, 2, 0, 3, 0, 3, 1, 2, 2, 0, 3, 1, 3, 1, 2, 0, 1, 2, 0, 3, 0, 3, 1, 2, 2, 0, 3, 1, 0, 3, 1, 2, 2, 0
Offset: 1

Views

Author

Clark Kimberling, May 31 2017

Keywords

Comments

This is the fixed point of the morphism 0->0231, 1->2310, 2->3102, 3->1023 starting with 0. Let t be the (nonperiodic) sequence of positions of 0, and likewise, u for 1, v for 2, and w for 3; then t(n)/n -> 4, u(n)/n -> 4, v(n)/n -> 4, w(n)/n -> 4. Also, t(n) + u(n) + v(n) + w(n) = 16*n - 6 for n >= 1. See A287556 for a guide to related sequences.

Examples

			First three iterations of the morphism:
0312
0312203131201203
0312203131201203120303122031312020313120120303123120120303122031

Links

Crossrefs

Cf. A287572, A287573, A287574, A287575.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {0,3,1,2}, 1 -> {3,1,2,0}, 2 -> {1,2,0,3}, 3 -> {2,0,3,1}}] &, {0}, 9];   (* A287571 *)
Flatten[Position[s, 0]]; (* A287572 *)
Flatten[Position[s, 1]]; (* A287573 *)
Flatten[Position[s, 2]]; (* A287574 *)
Flatten[Position[s, 3]]; (* A287575 *)

Formula

a(n) = 4n - A287574(n) for n >= 1.

A287576 Start with 0 and repeatedly substitute 0->0321, 1->3210, 2->2103, 3->1032.

Original entry on oeis.org

0, 3, 2, 1, 1, 0, 3, 2, 2, 1, 0, 3, 3, 2, 1, 0, 3, 2, 1, 0, 0, 3, 2, 1, 1, 0, 3, 2, 2, 1, 0, 3, 2, 1, 0, 3, 3, 2, 1, 0, 0, 3, 2, 1, 1, 0, 3, 2, 1, 0, 3, 2, 2, 1, 0, 3, 3, 2, 1, 0, 0, 3, 2, 1, 1, 0, 3, 2, 2, 1, 0, 3, 3, 2, 1, 0, 0, 3, 2, 1, 0, 3, 2, 1, 1, 0
Offset: 1

Views

Author

Clark Kimberling, Jun 01 2017

Keywords

Comments

This is the fixed point of the morphism 0->0231, 1->2310, 2->3102, 3->1023 starting with 0. Let t be the (nonperiodic) sequence of positions of 0, and likewise, u for 1, v for 2, and w for 3; then t(n)/n -> 4, u(n)/n -> 4, v(n)/n -> 4, w(n)/n -> 4. Also, t(n) + u(n) + v(n) + w(n) = 16*n - 6 for n >= 1. See A287556 for a guide to related sequences.

Examples

			First three iterations of the morphism:
0321
0321103221033210
0321103221033210321003211032210321033210032110321032210332100321

Links

Crossrefs

Cf. A287577, A287578, A287579, A287580.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {0,3,2,1}, 1 -> {3,2,1,0}, 2 -> {2,1,0,3}, 3 -> {1,0,3,2}}] &, {0}, 9];   (* A287576 *)
Flatten[Position[s, 0]]; (* A287577 *)
Flatten[Position[s, 1]]; (* A287578 *)
Flatten[Position[s, 2]]; (* A287579 *)
Flatten[Position[s, 3]]; (* A287580 *)

Formula

a(n) = 4n - A287578(n) for n >= 1.
Showing 1-8 of 8 results.

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