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A272729 a(n) is the number of repetitions of 2n-1 in A272727.

%I A272729 #18 Jun 26 2022 22:46:46
%S A272729 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 A272729 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 A272729 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 A272729 a(n) is the number of repetitions of 2n-1 in A272727.
%C A272729 Also, value of A272728 at the n-th local maximum.
%C A272729 Also, the trajectory of 1 under the morphism n->[1,1..1,n+1] (the number of 1's is n-1).
%C A272729 Average value tends to 2.
%C A272729 Number n makes its first appearance at the position 2^(n-1) and has frequency 1/2^n.
%C A272729 Conjectured first differences of A037988 (true for at least 8192 terms). - _Sean A. Irvine_, Jun 26 2022
%H A272729 Ivan Neretin, <a href="/A272729/b272729.txt">Table of n, a(n) for n = 1..8192</a>
%H A272729 <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>
%e A272729 The morphism acts as follows:
%e A272729 1->2; 2->1,3; 3->1,1,4; 4->1,1,1,5; etc.
%e A272729 The trajectory starts as:
%e A272729 1 ->
%e A272729 2 ->
%e A272729 1,3 ->
%e A272729 2,1,1,4 ->
%e A272729 1,3,2,2,1,1,1,5 -> ...
%e A272729 The result of k iterations is a series with 2^(k-1) terms; their sum is 2^k.
%e A272729 If A001511 is laid out in a similar irregular triangle, each row
%e A272729 would contain the same terms, albeit in a different order:
%e A272729 1,
%e A272729 2,
%e A272729 1,3,
%e A272729 1,2,1,4,
%e A272729 1,2,1,3,1,2,1,5...
%t A272729 Flatten@NestList[Flatten[Append[ConstantArray[1, # - 1], # + 1] & /@ #] &, {1}, 7]
%Y A272729 Cf. A001511, A272727, A272728.
%K A272729 nonn
%O A272729 1,2
%A A272729 _Ivan Neretin_, May 05 2016