Let's work on the finite field $\mathbb{Z}_2$ and define the matrix S as
\begin{equation} S = \begin{bmatrix} 1 & 0 & 0 & \ldots & 0 & 0 & 1 \\ 1 & 1 & 0 & \ldots & 0 & 0 & 0\\ 0 & 1 & 1 & \ldots & 0 & 0 & 0\\ \vdots & \vdots & \vdots &\ddots & \vdots \\ 0 & 0 & 0 & \ldots & 1 & 0 & 0 \\ 0 & 0 & 0 & \ldots & 1 & 1 & 0 \\ 0 & 0 & 0 & \ldots & 0 & 1 & 1 \\ \end{bmatrix} \end{equation}
The characteristic polynomial of this matrix (modulo 2) is
$$ p_n(\lambda) = (1 - \lambda)^n + 1. $$
This result can be easily proven by checking any matrix $S_{n\times n}$ and expressing its characteristic polynomial, $p_n(\lambda)$, in terms of the characteristic polynomials of its minors. Then, use induction.
Now, the question concerns the following matrix
\begin{equation} T = \begin{bmatrix} S & 0 & 0 & \ldots & 0 & 0 & S \\ S & S & 0 & \ldots & 0 & 0 & 0\\ 0 & S & S & \ldots & 0 & 0 & 0\\ \vdots & \vdots & \vdots &\ddots & \vdots \\ 0 & 0 & 0 & \ldots & S & 0 & 0 \\ 0 & 0 & 0 & \ldots & S & S & 0 \\ 0 & 0 & 0 & \ldots & 0 & S & S \\ \end{bmatrix}. \end{equation}
The matrix $T$ consists of blocks of $n\times n $ matrices, where it is assumed that all zero elements are in essence $0_{n\times n}$.
There are many ways to deal with the calculation of the characteristic polynomials of the matrix $T$. There are numerical solutions, as outlined in
What is the fastest way to find the characteristic polynomial of a matrix?
but also symbolic ones, by using Mathematica, Matlab, etc...
Finally there are arguments similar to the ones used for the calculation of the matrix $S$.
I want to calculate the characteristic polynomial of $T$ but there are some constraints:
I want to be able to reduce the resulting polynomial to its elementary divisors in the finite field $\mathbb{Z}_2$ and do another series of other algebraic operations on it (irrelevant to this problem). Thus, numerical efficiency matters.
I want to be able to calculate the characteristic polynomials (efficiently enough, meaning less than an hour, I guess...) up to sub-blocks of dimension $15\times 15$, at least. Preferably up to $40\times 40$.
From the above considerations, I tend to believe that the only efficient (and surely elegant) way to deal with the problem is to find a closed-form formula for $T$. I tried to follow the same logic as for $S$, but I don't see anything coming out...
The question is: is there a closed form solution for the characteristic polynomial of $T$? It can be in an infinite series form, a polynomial (as for $S$), whatever. Is there a solution in terms of elementary functions?
Note: Some partial solutions are known. E.g., for sub-blocks with power-of-2 dimension, the characteristic polynomial can be easily found to be $$ p_{2^n \times 2^n}(\lambda) = \lambda^{ 2^{2n} },\, with \;\; n\geq 1$$ where $2^n \times 2^n$ are the dimensions of each sub-block.
Some relevant bibliography (I would be very pleased if someone can provide other/more comprehensive works):
J. G. Stevens, R. E. Rosensweig, and A. E. Cerkanowicz, Transient and cyclic behavior of cellular automata with null boundary conditions, Journal of Statistical Physics 73, 159 (1993).
A. Ehrlich, Periods in Ducci’s n-number game of differences, Fibonacci Quarterly 28, 302 (1990).
N. J. Calkin, J. G. Stevens, and D. M. Thomas, A characterization for the length of cycles of the n-number Ducci game, Fibonacci Quarterly 43, 53 (2005).
