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A194688 First differences of A036554 (numbers whose binary representation ends in an odd number of zeros).

%I A194688 #24 Feb 15 2025 20:25:40
%S A194688 4,2,2,4,4,4,2,2,4,2,2,4,2,2,4,4,4,2,2,4,4,4,2,2,4,4,4,2,2,4,2,2,4,2,
%T A194688 2,4,4,4,2,2,4,2,2,4,2,2,4,4,4,2,2,4,2,2,4,2,2,4,4,4,2,2,4,4,4,2,2,4,
%U A194688 4,4,2,2,4,2,2,4,2,2,4,4,4,2,2,4,4,4,2,2,4,4,4,2,2,4,2,2,4,2,2,4,4,4,2,2,4,4,4,2,2,4,4,4,2,2,4,2,2,4,2,2,4,4,4,2,2,4,2,2,4,2,2,4,4,4,2,2,4,2
%N A194688 First differences of A036554 (numbers whose binary representation ends in an odd number of zeros).
%C A194688 Conjecture. This sequence is self-generated according to the following rule: start with {4} at step 0, then extend by steps, appending {2,2,4} at step n if a(n)=4 or appending {4} if a(n)=2. (This has been verified for several thousand terms.) To illustrate, the first few steps of this process give {4}->{4,2,2,4}, since a(1)=4, ->{4,2,2,4,4}, since a(2)=2, ->{4,2,2,4,4,4}, since a(3)=2, ->{4,2,2,4,4,4,2,2,4}, since a(4)=4, etc. Equivalently, it appears that {a(n)} is the fixed-point of the morphism 2->4, 4->422, starting with 4.
%C A194688 Since A036554 = 2*A003159, this conjecture follows from the paper by Allouche, Shallit and Skordev in 2005, see page 13. - _Michel Dekking_, Jan 06 2019
%C A194688 It appears that arbitrarily long runs of terms of this sequence occur in A023630 and A023632.
%H A194688 J.-P. Allouche, J. Shallit and G. Skordev, <a href="http://dx.doi.org/10.1016/j.disc.2004.12.004">Self-generating sets, integers with missing blocks and substitutions</a>, Discrete Math. 292 (2005) 1-15.
%t A194688 Differences[Select[Range[500],OddQ[IntegerExponent[#,2]]&]] (* _Harvey P. Dale_, Jun 29 2021 *)
%Y A194688 Cf. A003159, A036554.
%Y A194688 Cf. A023630, A023632.
%K A194688 nonn
%O A194688 1,1
%A A194688 _John W. Layman_, Sep 03 2011