The art of juggling with two balls or A proof for a modular condition of Lucas numbers

Abstract

In this short note we look at the problem of counting juggling patterns with one ball or two balls with a throw at every occurrence. We will do this for both traditional juggling and for spherical juggling. In the latter case we will show a connection to the "associated Mersenne numbers" (A001350) and so as a result will be able to recover a proof that the $p$th Lucas number is congruent to 1 modulo p when p is a prime.

4 pages; replaces previous version and adds references to similar existing proofs