A287564 -id:A287564 - cp's OEIS frontend

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

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

Clark Kimberling, May 31 2017

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.

			First three iterations of the morphism:
0213
0213130221303021
0213130221303021213030210213130213022130302102133021021313022130

Clark Kimberling, Table of n, a(n) for n = 1..20000
Index entries for sequences that are fixed points of mappings

Cf. A287562, A287563, A287564, A287565.

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 *)

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

A287562 Positions of 0 in A287561.

Original entry on oeis.org

1, 7, 12, 14, 20, 22, 25, 31, 35, 40, 42, 45, 50, 53, 59, 64, 67, 72, 74, 77, 82, 85, 91, 96, 97, 103, 108, 110, 116, 118, 121, 127, 132, 134, 137, 143, 147, 152, 154, 157, 162, 165, 171, 176, 177, 183, 188, 190, 194, 197, 203, 208, 209, 215, 220, 222, 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 3, 1, 0, 2, 0, 2, 3, 1, 1, 0, 2, 3, 2, 3, 1, 0, 1, 0, 2, 3, with
in positions given by A287563,
in positions given by A287564,
in positions given by A287565,
in positions given by A287562.

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

Cf. A287561, A287563, A287564, A287565.

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 *)

A287565 Positions of 3 in A287561.

Original entry on oeis.org

4, 6, 11, 13, 19, 21, 28, 30, 34, 39, 41, 48, 49, 56, 58, 63, 66, 71, 73, 80, 81, 88, 90, 95, 100, 102, 107, 109, 115, 117, 124, 126, 131, 133, 140, 142, 146, 151, 153, 160, 161, 168, 170, 175, 180, 182, 187, 189, 193, 200, 202, 207, 212, 214, 219, 221, 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 0, 2, 1, 3, 1, 3, 0, 2, 2, 1, 3, 0, 3, 0, 2, 1, 2, 1, 3, 0, with
in positions given by A287562,
in positions given by A287563,
in positions given by A287564,
in positions given by A287565.

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

Cf. A287561, A287562, A287563, A287565.

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 *)

a(n) = 4n - A287561(n) for n >= 1.

A287563 Positions of 1 in A287561.

Original entry on oeis.org

3, 5, 10, 16, 18, 24, 27, 29, 33, 38, 44, 47, 52, 55, 57, 62, 65, 70, 76, 79, 84, 87, 89, 94, 99, 101, 106, 112, 114, 120, 123, 125, 130, 136, 139, 141, 145, 150, 156, 159, 164, 167, 169, 174, 179, 181, 186, 192, 196, 199, 201, 206, 211, 213, 218, 224, 226

Offset: 1

Clark Kimberling, May 31 2017

a(n) - a(n-1) is in {1,2,3,4,5,6} 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, 2, 0, 2, 0, 1, 3, 3, 2, 0, 1, 0, 1, 3, 2, 3, 2, 0, 1, with
in positions given by A287565,
in positions given by A287562,
in positions given by A287563,
in positions given by A287564.

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

Cf. A287561, A287562, A287564, A287565.

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 *)

cp's OEIS Frontend

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

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A287562 Positions of 0 in A287561.

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A287565 Positions of 3 in A287561.

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A287563 Positions of 1 in A287561.

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