# Finite notation for natural numbers in base $(a+\sqrt{b})/2$

Natural numbers in base $$10$$ have a finite notation. But there are other bases of the form $$\frac{a+\sqrt b}{2}$$, where all natural numbers can be written in a finite notation. In base $$\frac{5+\sqrt 17}{2}$$ for example the decimal number $$11$$ would be written as $$21.4$$.

What is the condition for possible values $$(a,b)$$ so that natural numbers can be written in a finite notation ?