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

One more curious property.

Let

or

where the are the terms of the original sequence (1,1,2,2,4,4,6,6,10,10,...)

then

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:

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

One more curious property.

Let

or

where the are the terms of the original sequence (1,1,2,2,4,4,6,6,10,10,...)

then

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:

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

Gottfried Helms, Kassel