Let \(A:= \left( \begin{smallmatrix} 1&0\\ 1&1 \end{smallmatrix} \right)\), \(B:= \left( \begin{smallmatrix} 1&1\\ 0&1 \end{smallmatrix} \right)\), and for \(n\in \mathbb{N}\), let \(\Phi(n)\) be the number of matrices \(C\) which are products of \(A\)'s and \(B\)'s where both \(A\) and \(B\) must occur, such that the trace \(\text{Tr} (C)=n\). It had been conjectured that \(\Phi(n)\sim \frac{12}{\pi^2} n\log n\), \(n\to \infty\). This conjecture is disproved in the paper by explicitly showing that the quotient function \(\varphi(n):= \frac{\Phi(n)} {n\log n}\), \(n\geq 2\), has an absolutely continuous limit distribution with respect to the Lebesgue measure. In fact, the main result of this paper is the following: Theorem: The arithmetical function \(\varphi\) has a smooth limit distribution, i.e., there is some \(\delta\in C^\infty (0,\infty)\cap L^1(0,\infty)\) with \(\delta\geq 0\), \(\int_0^\infty \delta(t) dt= 1\), such that for \(0< a< b< \infty\), \(\lim_{x\to \infty} \frac 1x|\{n\leq x\mid \varphi(n)\in (a,b]\}|= \int_a^b \delta(t) dt\). Since according to this theorem the limit distribution of \(\varphi\) is absolutely continuous with respect to the Lebesgue measure, it turns out that the existing conjecture cannot hold, and at the same time it is replaced by this new positive statement. The essential part of the proof is to show that \(\varphi\) is 1-limit periodic.