The generalized Tur\'an number for K_3 in graphs without suspensions of a path on five vertices

Given graphs $H$ and $F$, the generalized Tur\'an number $\ex(n, H, F)$ is defined as the maximum number of copies of $H$ in an $n$-vertex graph that contains no copy of $F$. The suspension $\widehat{F}$ of a graph $F$ is obtained by adding a new vertex that is adjacent to every vertex of $F$. Mubayi and Mukherjee (2023, DM) conjectured that $\ex(n, K_3, \widehat{P_k})=\left\lfloor \frac{k-2}{2}\right\rfloor \cdot \frac{n^2}{8}+o(n^2)$, where $P_k$ is a path on $k\ge 4$ vertices. Using the triangle removal lemma, they verified this conjecture for $k=4,5,6$. Later, Mukherjee (2024, DM) established the exact value $\ex(n, K_3, \widehat{P_4})=\left\lfloor n^2/8\right\rfloor$. In this paper, using the stability method, we determine the exact value of $\ex(n, K_3, \widehat{P_5})$ by showing that for sufficiently large $n$, $\ex(n,K_3, \widehat{P_5})=\left\lfloor n^2/8\right\rfloor.$