Counting triangles in graphs without vertex disjoint odd cycles

Given two graphs $H$ and $F$, the maximum possible number of copies of $H$ in an $F$-free graph on $n$ vertices is denoted by $\mathrm{e}\mathrm{x}(n,H,F)$. Let $(\mathrm{\ell }+1)\cdot F$ denote $\mathrm{\ell }+1$ vertex disjoint copies of $F$. In this paper, we determine the exact value of $\mathrm{e}\mathrm{x}(n,C_3,(\mathrm{\ell }+1)\cdot C_{2k+1})$ and its extremal graph, which generalizes some known results.