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

1,1

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

Table of n, a(n) for n=1..138.

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

Mathematica

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

Crossrefs

Cf. A003159, A036554.

Cf. A023630, A023632.

Sequence in context: A170988 A141035 A100854 * A317389 A322510 A021707

Adjacent sequences: A194685 A194686 A194687 * A194689 A194690 A194691

Keyword

nonn

Author

John W. Layman, Sep 03 2011

Status

approved