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.

A287572 Positions of 0 in A287571.

Original entry on oeis.org

1, 6, 12, 15, 19, 21, 26, 32, 34, 40, 43, 45, 52, 55, 57, 62, 68, 71, 73, 78, 81, 86, 92, 95, 99, 101, 106, 112, 114, 120, 123, 125, 131, 133, 138, 144, 146, 152, 155, 157, 164, 167, 169, 174, 177, 182, 188, 191, 194, 200, 203, 205, 212, 215, 217, 222, 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, 2, 0, 1, 1, 3, 2, 0, 2, 0, 1, 3, 0, 1, 3, 2, 0, 1, 3, 2, with
0 in positions given by A287573,
1 in positions given by A287574,
2 in positions given by A287575,
3 in positions given by A287572.

Links

Crossrefs

Cf. A287571, 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 *)

A287575 Positions of 3 in A287571.

Original entry on oeis.org

2, 7, 9, 16, 20, 22, 27, 29, 35, 37, 44, 46, 49, 56, 58, 63, 65, 72, 74, 79, 82, 87, 89, 96, 100, 102, 107, 109, 115, 117, 124, 126, 132, 134, 139, 141, 147, 149, 156, 158, 161, 168, 170, 175, 178, 183, 185, 192, 195, 197, 204, 206, 209, 216, 218, 223, 226
Offset: 1

Views

Author

Clark Kimberling, Jun 01 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, 1, 3, 0, 0, 2, 1, 3, 1, 3, 0, 2, 3, 0, 2, 1, 3, 0, 2, 1, with
0 in positions given by A287574,
1 in positions given by A287575,
2 in positions given by A287572,
3 in positions given by A287573.

Links

Crossrefs

Cf. A287571, A287572, A287573, A287574.

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 *)

A287573 Positions of 1 in A287571.

Original entry on oeis.org

3, 8, 10, 13, 17, 23, 28, 30, 36, 38, 41, 47, 50, 53, 59, 64, 66, 69, 75, 80, 83, 88, 90, 93, 97, 103, 108, 110, 116, 118, 121, 127, 129, 135, 140, 142, 148, 150, 153, 159, 162, 165, 171, 176, 179, 184, 186, 189, 196, 198, 201, 207, 210, 213, 219, 224, 227
Offset: 1

Views

Author

Clark Kimberling, Jun 01 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, 0, 2, 3, 3, 1, 0, 2, 0, 2, 3, 1, 2, 3, 1, 0, 2, 3, 1, 0, with
0 in positions given by A287575,
1 in positions given by A287572,
2 in positions given by A287573,
3 in positions given by A287574.

Links

Crossrefs

Cf. A287571, A287572, 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 *)

A287574 Positions of 2 in A287571.

Original entry on oeis.org

4, 5, 11, 14, 18, 24, 25, 31, 33, 39, 42, 48, 51, 54, 60, 61, 67, 70, 76, 77, 84, 85, 91, 94, 98, 104, 105, 111, 113, 119, 122, 128, 130, 136, 137, 143, 145, 151, 154, 160, 163, 166, 172, 173, 180, 181, 187, 190, 193, 199, 202, 208, 211, 214, 220, 221, 228
Offset: 1

Views

Author

Clark Kimberling, Jun 01 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, 3, 1, 2, 2, 0, 3, 1, 3, 1, 2, 0, 1, 2, 0, 3, 1, 2, 0, 3, with
0 in positions given by A287572,
1 in positions given by A287573,
2 in positions given by A287574,
3 in positions given by A287575.

Links

Crossrefs

Cf. A287571, A287572, A287573, 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 - A287571(n) for n >= 1.

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 *)
Showing 1-5 of 5 results.

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