Number of words in the regular language $(00)^*$
For your language, can you take $p_0(x) = 1/2$, $\lambda_0 = 1$, $p_1(x) = 1/2$, $\lambda_1 = -1$, and $p_i(x) = \lambda_i = 0$ for $i &gt; 1$? The Wikipedia article doesn't say anything about the coefficients being either positive or integral. The sum for my choices is 

$\qquad \displaystyle 1/2 + 1/2(-1)^n = 1/2 (1 + (-1)^n)$ 

which seems to be 1 for even $n$, and 0 for odd $n$. Indeed, a proof by induction seems straightforward.