IMO there is neither a pentagon in the Fibonacci sequence nor a Fibonacci sequence in the pentagon. You may compute the $n^{th}$ Fibonacci number by $\displaystyle F_n=\left\lfloor\frac{1}{2}+\frac{\varphi^n}{\sqrt{5}} \right\rfloor$ what approximates for larger $n$ an almost constant ratio $F_n/F_{n+1}$. Taking this as a reason to surmise a relation with a pentagon is as if you try to square the circle me think.

There is neither a pentagon in the Fibonacci sequence nor a Fibonacci sequence in the pentagon. You may compute the $n^{th}$ Fibonacci number by $\displaystyle F_n=\left\lfloor\frac{1}{2}+\frac{\varphi^n}{\sqrt{5}} \right\rfloor$ what approximates for larger $n$ a converging ratio $F_n/F_{n+1}$. Taking this as a reason to surmise a relation with a pentagon is as if you try to square the circle.

Note: with the supplement of OP on 2024-07-09 21:00:04Z I see my point of view as affirmed.