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a(n) is the number of repetitions of 2n-1 in A272727.
8

%I #18 Jun 26 2022 22:46:46

%S 1,2,1,3,2,1,1,4,1,3,2,2,1,1,1,5,2,1,1,4,1,3,1,3,2,2,2,1,1,1,1,6,1,3,

%T 2,2,1,1,1,5,2,1,1,4,2,1,1,4,1,3,1,3,1,3,2,2,2,2,1,1,1,1,1,7,2,1,1,4,

%U 1,3,1,3,2,2,2,1,1,1,1,6,1,3,2,2,1,1,1,5,1,3,2,2,1,1,1,5

%N a(n) is the number of repetitions of 2n-1 in A272727.

%C Also, value of A272728 at the n-th local maximum.

%C Also, the trajectory of 1 under the morphism n->[1,1..1,n+1] (the number of 1's is n-1).

%C Average value tends to 2.

%C Number n makes its first appearance at the position 2^(n-1) and has frequency 1/2^n.

%C Conjectured first differences of A037988 (true for at least 8192 terms). - _Sean A. Irvine_, Jun 26 2022

%H Ivan Neretin, <a href="/A272729/b272729.txt">Table of n, a(n) for n = 1..8192</a>

%H <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>

%e The morphism acts as follows:

%e 1->2; 2->1,3; 3->1,1,4; 4->1,1,1,5; etc.

%e The trajectory starts as:

%e 1 ->

%e 2 ->

%e 1,3 ->

%e 2,1,1,4 ->

%e 1,3,2,2,1,1,1,5 -> ...

%e The result of k iterations is a series with 2^(k-1) terms; their sum is 2^k.

%e If A001511 is laid out in a similar irregular triangle, each row

%e would contain the same terms, albeit in a different order:

%e 1,

%e 2,

%e 1,3,

%e 1,2,1,4,

%e 1,2,1,3,1,2,1,5...

%t Flatten@NestList[Flatten[Append[ConstantArray[1, # - 1], # + 1] & /@ #] &, {1}, 7]

%Y Cf. A001511, A272727, A272728.

%K nonn

%O 1,2

%A _Ivan Neretin_, May 05 2016