Ternary Strings with No Consecutive $1^{\prime }$s

Abstract

For $n\in \mathbb{N}$, let $a_n$ count the number of ternary strings of length $n$ that contain no consecutive $1$s. We find that $a_n=(\frac{1}{2}+\frac{\sqrt{3}}{3}){(1+\sqrt{3})}^n–(\frac{1}{2}-\frac{\sqrt{3}}{3}){(1–\sqrt{3})}^n$. For a given $n\ge 0$, we then determine the following for these $a_n$ ternary strings:
(1)the number of $0^{\prime }$s, $1^{\prime }$s, and $2^{\prime }$s;(2)the number of runs;(3) the number of rises, levels, and descents; and
(4)the sum obtained when these strings are considered as base $3$ integers.

Following this, we consider the special case for those ternary strings (among the $a_n$ strings we first considered) that are palindromes, and determine formulas comparable to those in (1) – (4) above for this special case.