Let $b,n \geq 2$ be integers and let $\Phi_n(b)$ be the value of the $n$-th cyclotomic polynomial evaluated at $b$. I've recently noticed by computer experiments that whenever $n$ is odd, we seem to have the lower bound $$ \Phi_n(b) > \dfrac{b^n-1}{b^{\lceil n/2 \rceil} - 1 }. $$ Curiously this seems to be false whenever $n$ is even. I wonder why the disparity? Why is it that it works for $n$ odd but not for even $n$? Can we also prove the above lower bound? Thanks.
Update: Lucian has found the exceptions $n = 105$ and $n = 165$. The lower bound however still seems to work most of the time. The exceptions are interesting and one could ask whether there are a finite number of these or why they appear at all. But there is still no explanation as to why the bound never works for $n$ even, but works most of the time for $n$ odd.