  In this paper we establish some sophisticated congruences involving central
binomial coefficients and Fibonacci numbers. For example, we show that if
$p\not=2,5$ is a prime then
$$\sum_{k=0}^{p-1}F_{2k}\binom{2k}{k}=(-1)^{[p/5]}(1-(p/5)) (mod p^2)$$ and
$$\sum_{k=0}^{p-1}F_{2k+1}\binom{2k}k=(-1)^{[p/5]}(p/5) (mod p^2).$$ We also
obtain similar results for some other second-order recurrences and raise
several conjectures.
