A287579 -id:A287579

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A287576

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

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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.

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..20000

Index entries for sequences that are fixed points of mappings

FORMULA

a(n) = 4n - A287578(n) for n >= 1.

EXAMPLE

First three iterations of the morphism:

0321

0321103221033210

0321103221033210321003211032210321033210032110321032210332100321

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 *)

CROSSREFS

Cf. A287577, A287578, A287579, A287580.

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jun 01 2017

STATUS

approved

A287577

Positions of 0 in A287576.

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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

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

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 *)

CROSSREFS

Cf. A287576, A287578, A287579, A287580.

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jun 01 2017

STATUS

approved

A287580

Positions of 3 in A287576.

+10
6

2, 7, 12, 13, 17, 22, 27, 32, 36, 37, 42, 47, 51, 56, 57, 62, 67, 72, 73, 78, 82, 87, 92, 93, 97, 102, 107, 112, 116, 117, 122, 127, 132, 133, 138, 143, 147, 152, 153, 158, 162, 167, 172, 173, 177, 182, 187, 192, 193, 198, 203, 208, 212, 213, 218, 223, 227

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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 2, 1, 0, 3, 3, 2, 1, 0, 0, 3, 2, 1, 1, 0, 3, 2, 1, 0, 3, 2, with

0 in positions given by A287579,

1 in positions given by A287580,

2 in positions given by A287577,

3 in positions given by A287578.

LINKS

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

FORMULA

a(n) = 4n - A287576(n) for n >= 1.

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 *)

CROSSREFS

Cf. A287576, A287577, A287578, A287579.

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jun 01 2017

STATUS

approved

A287578

Positions of 1 in A287576.

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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

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

FORMULA

a(n) = 4n - A287576(n) for n >= 1.

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 *)

CROSSREFS

Cf. A287576, A287577, A287579, A287580.

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jun 01 2017

STATUS

approved

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