  The notion of $(a,b)$-cores is closely related to rational $(a,b)$ Dyck paths
due to Anderson's bijection, and thus the number of $(a,a+1)$-cores is given by
the Catalan number $C_a$. Recent research shows that $(a,a+1)$ cores with
distinct parts are enumerated by another important sequence- Fibonacci numbers
$F_a$. In this paper, we consider the abacus description of $(a,b)$-cores to
introduce the natural grading and generalize this result to $(a,as+1)$-cores.
We also use the bijection with Dyck paths to count the number of
$(2k-1,2k+1)$-cores with distinct parts. We give a second grading to Fibonacci
numbers, induced by bigraded Catalan sequence $C_{a,b} (q,t)$.
