It's hard to say if this is mere coincidence or part of a larger pattern. This is like asking someone to infer the next number to a finite sequence of given numbers. Whatever number you say, there is always some way to explain it.
Anyway, here's the "pattern" I see. Suppose $ A = \begin{pmatrix}B&u\\ v^T&\gamma\end{pmatrix}, $ where
- $B$ is a 2x2 upper triangular matrix;
- the two eigenvalues of $B$, say $\lambda$ and $\mu$, are distinct and $\neq\gamma$;
- $u$ is a right eigenvector of $B$ corresponding to the eigenvalue $\mu$;
- $v$ is a left eigenvector of $B$ corresponding to the eigenvalue $\lambda$.
Then $A$ has the following LU decomposition: $ A = \begin{pmatrix}B&u\\ v^T&\gamma\end{pmatrix} =\underbrace{\begin{pmatrix}I_2&0\\ kv^T&1\end{pmatrix}}_{L} \quad \underbrace{\begin{pmatrix}B&u\\0&\gamma\end{pmatrix}}_{U} $ where $k=\frac1\lambda$ if $\lambda\neq0$ or $0$ otherwise. The eigenvalues of $U$ are clearly $\lambda,\mu$ and $\gamma$. Since $u$ and $v$ are right and left eigenvectors of $B$ corresponding to different eigenvalues, we have $v^Tu=0$. Therefore \begin{align} (v^T, 0)A &=(v^T, 0)\begin{pmatrix}B&u\\ v^T&\gamma\end{pmatrix} =(v^TB,\, v^Tu)=\lambda(v^T,0),\\ A\begin{pmatrix}u\\0\end{pmatrix} &=\begin{pmatrix}B&u\\ v^T&\gamma\end{pmatrix}\begin{pmatrix}u\\0\end{pmatrix} =\begin{pmatrix}Bu\\v^Tu\end{pmatrix} =\mu\begin{pmatrix}u\\0\end{pmatrix},\\ A\begin{pmatrix}\frac{1}{\gamma-\mu}u\\1\end{pmatrix} &=\begin{pmatrix}B&u\\ v^T&\gamma\end{pmatrix} \begin{pmatrix}\frac{1}{\gamma-\mu}u\\1\end{pmatrix} =\gamma\begin{pmatrix}\frac{1}{\gamma-\mu}u\\1\end{pmatrix}. \end{align} So, the eigenvalues of $U$ are also the eigenvalues of $A$.