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)).

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

1,2

Comments

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.

Links

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

Rémy Sigrist, PARI program for A321695

Example

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.

Prog

(PARI) See Links section.

Crossrefs

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

Sequence in context: A174231 A072509 A269362 * A197432 A255573 A062298

Adjacent sequences: A321692 A321693 A321694 * A321696 A321697 A321698

Keyword

nonn

Author

Rémy Sigrist, Nov 17 2018

Status

approved