I am trying to find an asymptotic expansion for the following sum:
$$\sum_{n=1}^K \frac{\phi_0( 1/2+n+\lfloor(2n-1)/\sqrt{2}\rfloor)}{(4n-2)}$$
where $\phi_0$ is the digamma function and $\lfloor x\rfloor$ is the integer part of $x$. I found that, when $K$ tends to $\infty$, the result is
$$\displaystyle (\log^2{K})/8+\log(1+\sqrt{2}) (\log{K})/4+ 0.031914...$$
I would be very interested in knowing whether this constant term can be expressed in some other ways. I am not necessarily searching a strictly closed form, but rather a more elegant way to report it (e.g. as a function).