My favorite integer sequence

[Gottfried]

update: changed names of the functions to not to conflige with Jay's f(x)

---

One more curious property.
Let
\( \hspace{48} w(x)=1+1x+2x^2+2x^3+4x^4+4x^5+... \)
or
\( \hspace{48} w(x) = \sum_{k=0}^\infty a_k x^k \)
where the \( a_k \) are the terms of the original sequence (1,1,2,2,4,4,6,6,10,10,...)
then
\( \hspace{48} v(x)=1/w(x) = 1 - x -x^2 + x^3 -x^4 + x^5+x^6-x^7 \pm \ldots \)
is a much-discussed series (even in MSE and MO ( http://mathoverflow.net/questions/157326 ) ).
The pattern for the signs follow the Thue-Morse sequence (see wikipedia, or google "the ubiquituous thue morse sequence"), in letters: +--+ -++- -++- +--+ ... and has an interesting recursion:
\(
\hspace{48} A_1=+ \hspace{48} \hspace{48} B_1=- A_1 \\
\hspace{48} A_2=A_1 B_1, \hspace{48} B_2 = - A_2 \\
\hspace{48} A_3=A_2 B_2, \hspace{48} B_3 = - A_3 \\
\hspace{48} \cdots
\)


Hmm, I do not yet see a useful hint for the improvement of the computation of the terms of the sequence A nor for the taylor-series of Jay, but thought it might be additionally interesting.

Gottfried