Arithmetic properties of bipartitions with 3-cores

Recently, Lin discovered several nice congruences modulo 4, 5, 7 and 8 for A(3)(n), where A(3)(n) is the number of bipartitions with 3-cores of n. For example, Lin proved that for all alpha >= 0 and n >= 0, A(3)(16(alpha+1)n + 24(alpha+3)-2/3) = 0 (mod 5). Yao also established several infinite families of congruences modulo 3 and 9 for A(9)(n). In this paper, several infinite families of congruences modulo 4, 8 and 4(k)-1/3 (k >= 2) for A(3)(n) are established. We generalize some results due to Lin and Yao. For example, we prove that for n >= 0, a = 0 and k >= 2, A(3)(4(k(alpha+1))n + 2(2k(alpha+1)-1)-2/3) = 0 (mod 4(k)-1/3).