Here are some hints. Consider only the formulas $\displaystyle a_0,\dots, a_{2^n+1}$.

There are $\displaystyle 2^n$

valuations on n variables. For each valuation, how does the sequence $\displaystyle a_0,\dots, a_{2^n+1}$ of

*truth values* look like?

__Can we have $\displaystyle a_i=a_{i+2}\ne a_{i+1}$?__

Use pigeonhole principle where pigeons are pairs of formulas $\displaystyle (a_0,a_1), (a_1,a_2),\dots,(a_{2^n},a_{2^n+1})$ that can have different truth values for some valuation and holes are valuations.