  It is well known that a random cosine polynomial $ V_n(x) = \sum_ {j=0} ^{n}
a_j \cos (j x) , \ x \in (0,2 \pi) $, with the coefficients being independent
and identically distributed (i.i.d.) real-valued standard Gaussian random
variables (asymptotically) has $ 2n / \sqrt{3} $ expected real roots. On the
other hand, out of many ways to construct a dependent random polynomial, one is
to force the coefficients to be palindromic. Hence, it makes sense to ask how
many real zeros a random cosine polynomial (of degree $ n $) with identically
and normally distributed coefficients possesses if the coefficients are sorted
in palindromic blocks of a fixed length $ \ell. $ In this paper, we show that
the asymptotics of the expected number of real roots of such a polynomial is $
\mathrm{K}_\ell \cdot 2n / \sqrt{3} $, where the constant $ \mathrm{K}_\ell $
(depending only on $ \ell $) is greater than 1, and can be explicitly
represented by a double integral formula. That is to say, such polynomials have
slightly more expected real zeros compared with the classical case with i.i.d.
coefficients.
