%I #51 Feb 22 2020 20:54:24

%S 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,

%T 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,

%U 3,3,4,5,6,5,4,3,3,3,3,4,5,6,5,4,3,2,2

%N 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.

%C 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)-th run.

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

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

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

%C We can show that:

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

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

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

%C Conjectures:

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

%C 5) the sequence is unbounded.

%C 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.

%C This sequence has similarities with A113138. - _Rémy Sigrist_, Feb 08 2017

%H Rémy Sigrist, <a href="/A281579/b281579.txt">Table of n, a(n) for n = 1..10000</a>

%H Rémy Sigrist, <a href="/A281579/a281579.gp.txt">PARI program for A281579</a>

%e 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).

%e a(2)=2 also fits the definition.

%e (a(1), a(2)) constitutes the first run, and has length a(1)=2.

%e a(3) cannot equal 2 (as it would extend the previous run).

%e a(3)=3 fits the definition.

%e (a(2),a(3)) constitutes the second run, and has length a(2)=2.

%e a(4) cannot equal 2 (as a(5) would be equal to 1, which is impossible).

%e a(4)=3 fits the definition.

%e We complete the 3rd run with a(5)=3.

%Y Cf. A000002, A001462, A113138, A281772, A281783, A281900.

%K nonn

%O 1,1

%A _Rémy Sigrist_, Jan 29 2017