is every Series of the following form transcendental?

I've been fooling around with the Gregory series:

[latex]\sum_{k=1}^{\infty} \frac {(-1)^{k+1}}{2k-1}[/latex]

and brought it into the following form (since the series converges, I believe I can partition the Sum as I wish, I think. At least when I simulate it for [latex] n = 10^7[/latex] terms it still converges towards [latex]\frac {\pi}{4}[/latex]).

[latex]\frac{1}{2}\sum _{k=1}^n\frac{1}{\left(16k^2-16k+3\right)}[/latex]

Which sorta reminds me of the Geometric series, though it is quite different. My question is if the series of the following form is always transcendental:

[latex]\frac{1}{2}\sum _{k=1}^n\frac{1}{\left(16k^2-16k+w\right)}[/latex] where [latex]w \in \mathbb{N}[/latex]

besides for [latex] w = 3 [/latex]