On properties of solution of a reccurent equation appearing in enumeration of maximal independent sets in complete trees

The article considers a second-order nonlinear recurrent equation arising in analysis of the independent sets' quantity in complete $q$-ary trees. We proved earlier that for $q=2$ its solution has a limit and for any sufficiently large $q$ the solution splits into three converging subsequences with indices corresponding to the residue classes modulo 3. Computational experiment allowed to assume that this effect holds for any $q\geq 11$. The present paper proves divergence of the solution for any $q\geq 3$. The necessary condition for simultaneous convergence of all subsequences of the solution, with indices corresponding to the residue classes modulo 3, is the existence of a special solution of some nonlinear equations' system. Numerical search for solutions of the system, conducted in the present paper, showed that there is no corresponding solution of the system for any $3\leq q\leq 9$. We numerically and analytically show that the non-disintegrability into three subsequences takes place also for $q=10$.