A281579 Lexicographically earliest sequence such that, for any n>0, a(n)=length of the n-th run of consecutive terms in arithmetic progression in this sequence.

2, 2, 3, 3, 3, 4, 5, 4, 3, 3, 3, 3, 4, 5, 6, 7, 6, 5, 4, 4, 4, 3, 2, 2, 2, 3, 4, 4, 4, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 3, 3, 3, 3, 3, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 4, 4, 4, 5, 6, 7, 6, 5, 4, 4, 4, 2, 2, 3, 3, 3, 4, 5, 6, 5, 4, 3, 3, 3, 3, 4, 5, 6, 5, 4, 3, 2, 2

Offset: 1

Author

Rémy Sigrist, Jan 29 2017

Keywords

• nonn

Comments

Runs of consecutive terms in arithmetic progression overlap: the last term of the n-th run corresponds to the first term of the (n+1)-st run.

See A281772 for the common difference of the n-th run of consecutive terms in arithmetic progression.

See A281783 for the index of the first term of the n-th run of consecutive terms in arithmetic progression.

See A281900 for the index of the first occurrence of n in the sequence.

We can show that:

1) a(n)>=2 for any n>0,

2) a(n+1)<=a(n)+1 for any n>0,

3) runs of consecutive 2's have at least length 2.

Conjectures:

4) there are infinitely many runs of consecutive 2's,

5) the sequence is unbounded.

This sequence has connections with the Kolakoski sequence (A000002) and Golomb's sequence (A001462) in the sense that they all establish a link between their terms and the lengths of inner runs.

This sequence has similarities with A113138. - Rémy Sigrist, Feb 08 2017

A380317 is an essentially identical sequence. - N. J. A. Sloane, Feb 17 2025

Examples

			a(1)=2 fits the definition (and a(1)=1 would not, because whatever a(2) is, (a(1),a(2)) is an arithmetic progression of length 2).
a(2)=2 also fits the definition.
(a(1), a(2)) constitutes the first run, and has length a(1)=2.
a(3) cannot equal 2 (as it would extend the previous run).
a(3)=3 fits the definition.
(a(2),a(3)) constitutes the second run, and has length a(2)=2.
a(4) cannot equal 2 (as a(5) would be equal to 1, which is impossible).
a(4)=3 fits the definition.
We complete the 3rd run with a(5)=3.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, PARI program for A281579

Crossrefs

Cf. A000002, A001462, A113138, A281772, A281783, A281900, A380317.