A304588 -id:A304588 - cp's OEIS frontend

A338623 a(n) is the length of the longest block of consecutive terms appearing twice (possibly with overlap) among the first n terms of the Thue-Morse sequence (A010060).

0, 0, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 5, 6, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 11, 12, 13, 14, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25

Offset: 1

Rémy Sigrist, Nov 04 2020

nonn

This sequence is weakly increasing and unbounded (as A010060 is bounded).

			For n = 8:
- the first 8 terms of A010060 are 0, 1, 1, 0, 1, 0, 0, 1,
- the blocks (0, 1) and (1, 0) are repeated, and there are no length-3 repeated block,
- so a(8) = 2.

Rémy Sigrist, Table of n, a(n) for n = 1..8192
Rémy Sigrist, PARI program for A338623

Cf. A010060, A304588, A338624 (Kolakoski variant).

PARI
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See Links section.
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a(n) <= a(n+1) <= a(n) + 1.

a(A304588(n)) = n (and this is the first occurrence of n in the sequence).

A338626 a(n) is the least k such that the first k terms of the Kolakoski sequence (A000002) contain a length-n repeated block.

Original entry on oeis.org

3, 6, 9, 10, 13, 15, 16, 28, 32, 33, 40, 41, 42, 43, 69, 70, 71, 72, 94, 95, 96, 97, 98, 99, 106, 107, 108, 109, 214, 215, 216, 217, 218, 219, 220, 221, 222, 297, 298, 299, 300, 301, 339, 340, 487, 719, 720, 721, 722, 723, 724, 725, 726, 727, 728, 729, 730

Offset: 1

Rémy Sigrist, Nov 04 2020

nonn

The sequence contains large runs of consecutive integers.

			For n = 2:
- the first 5 terms of Kolakoski sequence are: 1, 2, 2, 1, 1,
- they have no length-2 repeated block,
- the first 6 terms of Kolakoski sequence are: 1, 2, 2, 1, 1, 2,
- they have a length-2 repeated block: 1, 2,
- so a(2) = 6.

Rémy Sigrist, Table of n, a(n) for n = 1..9511 (terms < 3000000)
Rémy Sigrist, PARI program for A338626
Rémy Sigrist, Perl program for A338626

Cf. A000002, A304588 (Thue-Morse variant), A338624.

PARI
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See Links section.
```
Perl
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See Links section.
```

A338624(a(n)) = n.

cp's OEIS Frontend

A338623 a(n) is the length of the longest block of consecutive terms appearing twice (possibly with overlap) among the first n terms of the Thue-Morse sequence (A010060).

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