This seems to be a question that does not have a really satisfactory answer yet--at least to my knowledge--perhaps that is why nobody tried to answer it yet.
Here is a not very satisfactory (because not really explicit) answer.
Take a nonzero element $a$ in the cokernel of the stable J homomorfism. Choose a framed manifold representing an element in the stable homotopy groups of spheres, belonging to $a$. If the dimension of this manifold is odd, then by framed surgery one can kill all the homotopy groups below its dimension, and then you obtain an element in $\Theta_n -bP_{n+1}.$

This construction uses the isomorphism
$Coker J \approx \Theta_n/bP_{n+1}.$

Here $J$ is the stable $J$-homomorphism $\pi_n(O) \to \pi_s(n).$