Summary: We study generalizations of the sequence of the $n$-anacci constants that are constructed from the ratio limits generated by linear recurrences of an arbitrary order $n$ with equal integer weights $m$. We derive the analytic representation of the class $C^{\infty }$ of these ratio limits and prove that, for a fixed $m$, the ratio limits form a strictly increasing sequence converging to $m+1$. We also show that the generalized $n$-anacci constants form a totally ordered set.

11B37, 11B39

linear recurrence, n-step Fibonacci number, weighted n-generalized Fibonacci sequence, generalized n-anacci constant