We successfully extend the standard Fibonacci zone plates with two on-axis foci to the generalized Fibonacci photon sieves (GFiPS) with multiple on-axis foci. We also propose the direct and inverse design methods based on the characteristic roots of the recursion relation of the generalized Fibonacci sequences. By switching the transparent and opaque zones, according to the generalized Fibonacci sequences, we not only realize adjustable multifocal distances but also fulfill the adjustable compression ratio of focal spots in different directions.