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

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, 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, 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

Offset: 1

Author

John W. Layman, Sep 03 2011

Keywords

• nonn

Comments

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.

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

It appears that arbitrarily long runs of terms of this sequence occur in A023630 and A023632.

Links

• J.-P. Allouche, J. Shallit and G. Skordev, Self-generating sets, integers with missing blocks and substitutions, Discrete Math. 292 (2005) 1-15.

Crossrefs

Cf. A003159, A036554.

Cf. A023630, A023632.

Programs

Mathematica

Differences[Select[Range[500],OddQ[IntegerExponent[#,2]]&]] (* Harvey P. Dale, Jun 29 2021 *)