  Second order recurrence relations of real numbers arise form various
applications in discrete time dynamical systems as well as in the context on
Markov chains. Solutions to the recurrence relations are fully defined by the
first two initial values as well as the recurrence formula. We calculate in
this work explicitly as a function of $a_1$ the minimal positive solution
$(a_i)_{i\in\mathbb{N}}$ to non-homogeneous second order recurrence relation
with affine coefficients when the non-homogeneity is constant and negative, and
the first initial value equals $a_0=0$. We show that rational coefficients lead
to a sequence of transcendental numbers. Additionally, we prove that this
sequence is the only bounded solution when varying $a_1$, converges to $0$ and
obtain the convergence speed in $O(i^{-1})$. We comment in the last section
further on the choice of rational parameters in the recurrence relation and we
make a link to the impossibility of obtaining computer based visualizations of
the minimal positive solution $(a_i)_{i\in\mathbb{N}}$.
