Exact generalized Turán number for $K_3$ versus suspension of $P_4$

Let $P_4$ denote the path graph on $4$ vertices. The suspension of $P_4$, denoted by ${\widehat{P}}_4$, is the graph obtained via adding an extra vertex and joining it to all four vertices of $P_4$. In this note, we demonstrate that for $n\ge 8$, the maximum number of triangles in any $n$-vertex graph not containing ${\widehat{P}}_4$ is $\lfloor n^2\mathrm{/}8\rfloor$. Our method uses simple induction along with computer programming to prove a base case of the induction hypothesis.