A Parity Result for Some p -Regular Partitions

Let p > 2 be prime and r ∈ { 1 , 2 , … , p − 1 } . Denote by c p ( n ) the number of p -regular partitions of n in which parts can occur not more than three times. We prove the following: If 8 r + 1 is a quadratic non-residue modulo p , c p ( p n + r ) ≡ 0 ( mod 2 ) for all nonnegative integers n .