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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

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

0 in positions given by A287559,

1 in positions given by A287560,

2 in positions given by A287557,

3 in positions given by A287558.

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

0 in positions given by A287557,

1 in positions given by A287558,

2 in positions given by A287560,

3 in positions given by A287559.

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

0 in positions given by A287560,

1 in positions given by A287557,

2 in positions given by A287558,

3 in positions given by A287559.