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 1-5 of 5 results.

A287556 Start with 0 and repeatedly substitute 0->0132, 1->1320, 2->3201, 3->2013.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, May 31 2017

Keywords

Comments

This is the fixed point of the morphism 0->0132, 1->1320, 2->3201, 3->2013 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.
In the following guide to related sequences, column 1 indexes fixed points on {0,1,2,3}, and column 2 indicates position sequences of 0, 1, 2, 3. Those sequences therefore comprise a 4-way splitting of the positive integers.
Fixed points of morphisms: Position sequences:
A053839: 0->0123, 1->1230, 2->2301, 3->3012 A287552-A287555
A287556: 0->0132, 1->1320, 2->3201, 3->2013 A287557-A287560
A287561: 0->0213, 1->2130, 2->1302, 3->3021 A287562-A287565
A287566: 0->0231, 1->2310, 2->3102, 3->1023 A287567-A287570
A287571: 0->0312, 1->3120, 2->1203, 3->2031 A287572-A287575
A287576: 0->0321, 1->3210, 2->2103, 3->1032 A287577-A287580

Examples

			First three iterations of the morphism:
0132
0132132020133201
0132132020133201132020133201013232010132132020132013320101321320

Links

Crossrefs

Cf. A287557, 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 *)

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.

A287577 Positions of 0 in A287576.

Original entry on oeis.org

1, 6, 11, 16, 20, 21, 26, 31, 35, 40, 41, 46, 50, 55, 60, 61, 66, 71, 76, 77, 81, 86, 91, 96, 100, 101, 106, 111, 115, 120, 121, 126, 131, 136, 137, 142, 146, 151, 156, 157, 161, 166, 171, 176, 180, 181, 186, 191, 196, 197, 202, 207, 211, 216, 217, 222, 226
Offset: 1

Views

Author

Clark Kimberling, Jun 01 2017

Keywords

Comments

a(n) - a(n-1) is in {1,4,5} 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, 2, 1, 0, 0, 3, 2, 1, 1, 0, 3, 2, 2, 1, 0, 3, 2, 1, 0, 3, with
0 in positions given by A287578,
1 in positions given by A287579,
2 in positions given by A287580,
3 in positions given by A287577.

Links

Crossrefs

Cf. A287576, 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 *)

A287578 Positions of 1 in A287576.

Original entry on oeis.org

4, 5, 10, 15, 19, 24, 25, 30, 34, 39, 44, 45, 49, 54, 59, 64, 65, 70, 75, 80, 84, 85, 90, 95, 99, 104, 105, 110, 114, 119, 124, 125, 130, 135, 140, 141, 145, 150, 155, 160, 164, 165, 170, 175, 179, 184, 185, 190, 195, 200, 201, 206, 210, 215, 220, 221, 225
Offset: 1

Views

Author

Clark Kimberling, Jun 01 2017

Keywords

Comments

a(n) - a(n-1) is in {1,4,5} 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, 3, 2, 1, 1, 0, 3, 2, 2, 1, 0, 3, 3, 2, 1, 0, 3, 2, 1, 0, with
0 in positions given by A287577,
1 in positions given by A287578,
2 in positions given by A287579,
3 in positions given by A287580.

Links

Crossrefs

Cf. A287576, A287577, 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 - A287576(n) for n >= 1.

A287579 Positions of 2 in A287576.

Original entry on oeis.org

3, 8, 9, 14, 18, 23, 28, 29, 33, 38, 43, 48, 52, 53, 58, 63, 68, 69, 74, 79, 83, 88, 89, 94, 98, 103, 108, 109, 113, 118, 123, 128, 129, 134, 139, 144, 148, 149, 154, 159, 163, 168, 169, 174, 178, 183, 188, 189, 194, 199, 204, 205, 209, 214, 219, 224, 228
Offset: 1

Views

Author

Clark Kimberling, Jun 01 2017

Keywords

Comments

a(n) - a(n-1) is in {1,4,5} 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, 0, 3, 2, 2, 1, 0, 3, 3, 2, 1, 0, 0, 3, 2, 1, 0, 3, 2, 1, with
0 in positions given by A287580,
1 in positions given by A287577,
2 in positions given by A287578,
3 in positions given by A287579.

Links

Crossrefs

Cf. A287576, A287577, A287578, 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 - A287576(n) for n >= 1.
Showing 1-5 of 5 results.

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