Find all such $b\in\mathbb{Z}_+$ that for all $n\in\mathbb{Z}_+$ there exists $y\in\mathbb{Z}_+$ such that $b^n+1=2y^3$

Question

The question is how to find all such positive integer $b\in\mathbb{Z}_+$ that for any positive integer $n\in\mathbb{Z}_+$ there exists a positive integer $y\in\mathbb{Z}_+$ such that the following equality stands: $$b^n+1=2y^3$$

Well, I tried $b=1$ and it fits. Also, I figured that $b$ must be odd :)