Arithmetic properties for Fu's 9 dots bracelet partitions

The notion of Fu's k dots bracelet partitions was introduced by Shishuo Fu. For any positive integer k, let B-k(n) denote the number of Fu's k dots bracelet partitions of n. Fu also proved several congruences modulo primes and modulo powers of 2. Recently, Radu and Sellers extended the set of congruences proven by Fu by proving three congruences modulo squares of primes for B-5(n), B-7(n) and B-11(n). More recently, Cui and Gu, and Xia and the author derived a number of congruences modulo powers of 2 for B-5(n). In this paper, we prove four congruences modulo 2 and two congruences modulo 4 for B-9(n) by establishing the generating functions of B-9(An + B) modulo 4 for some values of A and B.