>>391-393
ありがとう
スレ主です

１）log(2)の展開を使うスジの類似が下記で
　log 4/πの展開を使っている （こっちが本式かも）
　なお >>374 でも 自然対数ln(4/π)の展開と対比しているね https://arxiv.org/abs/math/0508042
２）「半分のn」を使うスジは、下記の式(3)だね

勉強になりました

（参考）
https://arxiv.org/pdf/0808.0410.pdf
VACCA-TYPE SERIES FOR VALUES OF THE
GENERALIZED-EULER-CONSTANT FUNCTION AND ITS DERIVATIVE
KH. HESSAMI PILEHROOD AND T. HESSAMI PILEHROOD
1. Introduction
In [11], J. Sondow proved the following two formulas:
(1) γ＝ 略
(2) log 4/π＝ 略
where γ is Euler’s constant and Ni,2(n) is the number of i’s in the binary expansion of n.
The series (1) is equivalent to the well-known Vacca series [13]
(3) γ＝ 略
and both series (1) and (3) may be derived from Catalan’s integral
(4) γ＝ 略
To see this it suffices to note tha
　G(x)＝ 略
is a generating function of the sequence N1,2(n)+N0,2(n), (see [10, sequence A070939]),
which is the binary length of n, rewrite (4) as