This time, I want to approach the connection between graphs and recursion relations from the point of view of matrices.

Given a recurrence relation with non-negative coefficients $$a_{n+m}=\sum_{j=1}^m\beta_ja_{n+m-j},$$ we can associate with it an m x m matrix

$$\left(\begin{array}{c}a_{n+m}\\\\a_{n+m-1}\\\\a_{n+m-2}\\\\\vdots\\\\a_{n+1}\end{array}\right)=\left(\begin{array}{cccc}\beta_1 & \beta_2&\cdots&\beta_m\\\\1&0&\cdots&0\\\\0&1&\cdots&0\\\\\vdots&\ddots&&\vdots\\\\0&\cdots&1&0\end{array}\right)\left(\begin{array}{c}a_{n+m-1}\\\\a_{n+m-2}\\\\a_{n+m-3}\\\\\vdots\\\\a_n\end{array}\right)$$ with non-negative entries. If we interpret this matrix as an adjacency matrix, its corresponding graph is the one I described in [comment #69](https://forum.azimuthproject.org/discussion/comment/18969/#Comment_18969).

Given a recurrence relation with non-negative coefficients $$a_{n+m}=\sum_{j=1}^m\beta_ja_{n+m-j},$$ we can associate with it an m x m matrix

$$\left(\begin{array}{c}a_{n+m}\\\\a_{n+m-1}\\\\a_{n+m-2}\\\\\vdots\\\\a_{n+1}\end{array}\right)=\left(\begin{array}{cccc}\beta_1 & \beta_2&\cdots&\beta_m\\\\1&0&\cdots&0\\\\0&1&\cdots&0\\\\\vdots&\ddots&&\vdots\\\\0&\cdots&1&0\end{array}\right)\left(\begin{array}{c}a_{n+m-1}\\\\a_{n+m-2}\\\\a_{n+m-3}\\\\\vdots\\\\a_n\end{array}\right)$$ with non-negative entries. If we interpret this matrix as an adjacency matrix, its corresponding graph is the one I described in [comment #69](https://forum.azimuthproject.org/discussion/comment/18969/#Comment_18969).