Suppose $ \left(a+\dfrac12\right)^n+\left(b+\dfrac12\right)^n=\dfrac{(2a+1)^n+(2b+1)^n}{2^n}\in\mathbb{Z}$. Let $ 2a+1=A$, $ 2b+1=B$. If $ n\ge2$, then $ n$ must be odd, otherwise $ A^n+B^n\equiv2\pmod 4$. Then $ A^n+B^n=(A+B)(A^{n-1}-A^{n-2}B+A^{n-3}B^2-\ldots-AB^{n-2}+B^{n-1})$. The second paranthesis consists of an odd ($ n$) number of odd terms, so it is odd. Then $ 2^n|A+B$, which holds for finitely many $ n$.