  In this paper, we propose an $(s+d,d)$-abacus for $(s,s+d,\dots,s+pd)$-core
partitions and establish a bijection between the $(s,s+d,\dots,s+pd)$-core
partitions and the rational Motzkin paths of type $(s+d,-d)$. This result not
only gives a lattice path interpretation of the $(s,s+d,\dots,s+pd)$-core
partitions but also counts them with a closed formula. Also we enumerate
$(s,s+1,\dots,s+p)$-core partitions with $k$ corners and self-conjugate
$(s,s+1,\dots,s+p)$-core partitions.
