A287568 -id:A287568 - cp's OEIS frontend

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

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

Clark Kimberling, May 31 2017

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.

			First three iterations of the morphism:
0231
0231310210232310
0231310210232310102323100231310223100231310210233102102323100231

Index entries for sequences that are fixed points of mappings

Cf. A287567, A287568, A287569, A287570.

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 *)

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

A287567 Positions of 0 in A287566.

Original entry on oeis.org

1, 7, 10, 16, 18, 24, 25, 31, 36, 37, 43, 46, 51, 54, 60, 61, 68, 69, 75, 78, 83, 86, 92, 93, 97, 103, 106, 112, 114, 120, 121, 127, 131, 134, 140, 141, 145, 151, 154, 160, 162, 168, 169, 175, 180, 181, 187, 190, 194, 200, 201, 207, 212, 213, 219, 222, 227

Offset: 1

Clark Kimberling, May 31 2017

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, 1, 2, 0, 2, 0, 3, 1, 0, 3, 1, 2, 1, 2, 0, 3, 0, 3, 1, 2, with
in positions given by A287568,
in positions given by A287569,
in positions given by A287570,
in positions given by A287567.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A287566, A287568, A287569, A287570.

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 *)

A287570 Positions of 3 in A287566.

Original entry on oeis.org

3, 5, 12, 14, 20, 22, 27, 29, 34, 39, 41, 48, 49, 56, 58, 63, 66, 71, 73, 80, 81, 88, 90, 95, 99, 101, 108, 110, 116, 118, 123, 125, 129, 136, 138, 143, 147, 149, 156, 158, 164, 166, 171, 173, 178, 183, 185, 192, 196, 198, 203, 205, 210, 215, 217, 224, 225

Offset: 1

Clark Kimberling, May 31 2017

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, 3, 0, 2, 0, 2, 1, 3, 2, 1, 3, 0, 3, 0, 2, 1, 2, 1, 3, 0, with
in positions given by A287570,
in positions given by A287567,
in positions given by A287568,
in positions given by A287569.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A287566, A287567, A287568, A287569.

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 *)

A287569 Positions of 2 in A287566.

Original entry on oeis.org

2, 8, 11, 13, 19, 21, 26, 32, 33, 38, 44, 47, 52, 55, 57, 62, 65, 70, 76, 79, 84, 87, 89, 94, 98, 104, 107, 109, 115, 117, 122, 128, 132, 135, 137, 142, 146, 152, 155, 157, 163, 165, 170, 176, 177, 182, 188, 191, 195, 197, 202, 208, 209, 214, 220, 223, 228

Offset: 1

Clark Kimberling, May 31 2017

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, 0, 1, 3, 1, 3, 2, 0, 3, 2, 0, 1, 0, 1, 3, 2, 3, 2, 0, 1, with
in positions given by A287569,
in positions given by A287570,
in positions given by A287567,
in positions given by A287568.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A287566, A287567, A287568, A287570.

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 *)

cp's OEIS Frontend

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

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A287567 Positions of 0 in A287566.

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A287570 Positions of 3 in A287566.

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A287569 Positions of 2 in A287566.

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