A288723 -id:A288723 - cp's OEIS frontend

A288724 Second sequence of a Kolakoski 3-Ouroboros, i.e., sequence of 1s, 2s and 3s that is second in a chain of three distinct sequences where successive run-length encodings produce seq(1) -> seq(2) -> seq(3) -> seq(1).

2, 2, 2, 3, 1, 1, 2, 2, 3, 3, 3, 1, 1, 1, 2, 3, 1, 2, 2, 3, 3, 1, 1, 2, 2, 2, 3, 1, 1, 2, 2, 2, 3, 3, 3, 1, 2, 3, 1, 1, 2, 2, 2, 3, 1, 2, 2, 3, 3, 1, 1, 1, 2, 2, 2, 3, 3, 3, 1, 2, 3, 1, 1, 2, 2, 3, 3, 1, 1, 1, 2, 3, 3, 1, 1, 1, 2, 2, 2, 3, 1, 2, 2, 3, 3, 1, 1

Offset: 1

Anthony Sand, Jun 14 2017

nonn

See comments at A288723.

			Write down the run-lengths of the sequence A288723, or the lengths of the runs of 1s, 2s and 3s. This yields a second and different sequence of 1s, 2s and 3s, A288724 (as above). The run-lengths of this second sequence yield a third and different sequence, A288725. The run-lengths of this third sequence yield the original sequence. For example, bracket the runs of distinct integers, then replace the original digits with the run-lengths to create the second sequence:
(1,1), (2,2), (3,3), (1,1,1), (2), (3), (1,1), (2,2), (3,3,3), (1,1,1), (2,2,2), (3), (1), (2), (3,3), (1,1,1), (2), (3,3), (1,1), (2,2,2), ... -> 2, 2, 2, 3, 1, 1, 2, 2, 3, 3, 3, 1, 1, 1, 2, 3, 1, 2, 2, 3, ...
Apply the same process to the second sequence and the third sequence appears:
(2,2,2), (3), (1,1), (2,2), (3,3,3), (1,1,1), (2), (3), (1), (2,2), (3,3), (1,1), (2,2,2), (3), (1,1), (2,2,2), (3,3,3), (1), (2), (3), ... -> 3, 1, 2, 2, 3, 3, 1, 1, 1, 2, 2, 2, 3, 1, 2, 3, 3, 1, 1, 1, ...
Apply the same process to the third sequence and the original sequence reappears:
(3), (1), (2,2), (3,3), (1,1,1), (2,2,2), (3), (1), (2), (3,3), (1,1,1), (2), (3), (1,1), (2,2), (3,3,3), (1,1,1), (2,2,2), (3), (1), ... -> 1, 1, 2, 2, 3, 3, 1, 1, 1, 2, 3, 1, 1, 2, 2, 3, 3, 3, 1, 1, ...

Georg Fischer, Table of n, a(n) for n = 1..2000 (recovered b-file, Jan 16 2019)
Rémy Sigrist, PARI program for A288723, A288724 and A288725

Cf. A000002, A025142, A025143. A288723 and A288725 are the first and third sequences in this 3-Ouroboros.

PARI
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See Links section.
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Data corrected by Rémy Sigrist, Oct 07 2017

A288725 Third sequence of a Kolakoski 3-Ouroboros, i.e., sequence of 1s, 2s and 3s that is third in a chain of three distinct sequences where successive run-length encodings produce seq(1) -> seq(2) -> seq(3) -> seq(1).

Original entry on oeis.org

3, 1, 2, 2, 3, 3, 1, 1, 1, 2, 2, 2, 3, 1, 2, 3, 3, 1, 1, 1, 2, 3, 1, 1, 2, 2, 3, 3, 3, 1, 1, 1, 2, 2, 2, 3, 1, 2, 3, 3, 1, 1, 2, 2, 3, 3, 3, 1, 2, 2, 3, 3, 3, 1, 1, 1, 2, 3, 1, 2, 2, 3, 3, 3, 1, 1, 1, 2, 3, 1, 1, 2, 2, 3, 3, 1, 1, 1, 2, 2, 2, 3, 3, 3, 1, 2, 2

Offset: 1

Anthony Sand, Jun 14 2017

nonn

See comments at A288723.

			Write down the run-lengths of the sequence A288723, or the lengths of the runs of 1s, 2s and 3s. This yields a second and different sequence of 1s, 2s and 3s, A288724. The run-lengths of this second sequence yield a third and different sequence, A288725 (as above). The run-lengths of this third sequence yield the original sequence. For example, bracket the runs of distinct integers, then replace the original digits with the run-lengths to create the second sequence:
(1,1), (2,2), (3,3), (1,1,1), (2), (3), (1,1), (2,2), (3,3,3), (1,1,1), (2,2,2), (3), (1), (2), (3,3), (1,1,1), (2), (3,3), (1,1), (2,2,2), ... -> 2, 2, 2, 3, 1, 1, 2, 2, 3, 3, 3, 1, 1, 1, 2, 3, 1, 2, 2, 3, ...
Apply the same process to the second sequence and the third sequence appears:
(2,2,2), (3), (1,1), (2,2), (3,3,3), (1,1,1), (2), (3), (1), (2,2), (3,3), (1,1), (2,2,2), (3), (1,1), (2,2,2), (3,3,3), (1), (2), (3), ... -> 3, 1, 2, 2, 3, 3, 1, 1, 1, 2, 2, 2, 3, 1, 2, 3, 3, 1, 1, 1, ...
Apply the same process to the third sequence and the original sequence reappears:
(3), (1), (2,2), (3,3), (1,1,1), (2,2,2), (3), (1), (2), (3,3), (1,1,1), (2), (3), (1,1), (2,2), (3,3,3), (1,1,1), (2,2,2), (3), (1), ... -> 1, 1, 2, 2, 3, 3, 1, 1, 1, 2, 3, 1, 1, 2, 2, 3, 3, 3, 1, 1, ...

Georg Fischer, Table of n, a(n) for n = 1..2000 (recovered b-file, Jan 16 2019)

Cf. A000002, A025142, A025143. A288723 and A288724 are the first and second sequences in this 3-Ouroboros.

A321695 For any sequence f of positive integers, let g(f) be the unique Golomb-like sequence with run lengths given by f and let k(f) be the unique Kolakoski-like sequence with run lengths given by f and initial term 1; this sequence is the unique sequence f satisfying f = g(k(f)).

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 5, 6, 6, 7, 7, 8, 8, 9, 10, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 17, 18, 19, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 28, 29, 30, 31, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 38, 38, 39, 39, 40, 41, 42, 43, 44, 45, 46, 47

Offset: 1

Rémy Sigrist, Nov 17 2018

nonn

More precisely:
- g(f) is the lexicographically earliest nondecreasing sequence of positive numbers whose RUNS transform equals f,
- k(f) is the lexicographically earliest sequence of 1's and 2's whose RUNS transform equals f,
- in particular:
     g(A001462) = A001462,
     k(A000002) = A000002,
     A321020 = g(A001462).
See A321696 for the RUNS transform of this sequence.
By applying twice the RUNS transform on this sequence, we recover the initial sequence; the same applies for A321696.
This sequence has connections with A288723; in both cases, we have sequences cyclically connected by RUNS transforms.

			We can build this sequence alongside A321696 iteratively:
- this sequence starts with 1,
- hence A321696 starts with 1, 2 (after the initial run of 1's, we have a run of 2's),
- hence this sequence starts with 1, 2, 2, 3 (after the runs of 1's and 2's, we have a run of 3's),
- hence A321696 starts with 1, 2, 2, 1, 1, 2, 2, 2, 1,
- hence this sequence starts 1, 2, 2, 3, 3, 4, 5, 6, 6, 7, 7, 8, 8, 9, 10,
- etc.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A321695

Cf. A000002, A001462, A288723, A321020, A321696.

PARI
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See Links section.
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cp's OEIS Frontend

A288724 Second sequence of a Kolakoski 3-Ouroboros, i.e., sequence of 1s, 2s and 3s that is second in a chain of three distinct sequences where successive run-length encodings produce seq(1) -> seq(2) -> seq(3) -> seq(1).

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A288725 Third sequence of a Kolakoski 3-Ouroboros, i.e., sequence of 1s, 2s and 3s that is third in a chain of three distinct sequences where successive run-length encodings produce seq(1) -> seq(2) -> seq(3) -> seq(1).

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Author

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Comments

Examples

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Crossrefs

A321695 For any sequence f of positive integers, let g(f) be the unique Golomb-like sequence with run lengths given by f and let k(f) be the unique Kolakoski-like sequence with run lengths given by f and initial term 1; this sequence is the unique sequence f satisfying f = g(k(f)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI