Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  prove that the property of being closed (resp., palindromic, rich, privileged trapezoidal, balanced) is expressible in first-order logic for automatic (and some related) sequences. It therefore follows that the characteristic function of those $n$ for which an automatic sequence $\bf x$ has a closed (resp., palindromic, privileged, rich, trapezoidal, balanced) factor of length  $n$ is itself automatic. For privileged words this requires a new characterization of the privileged property.  compute the corresponding characteristic functions for various famous sequences, such as the Thue-Morse sequence, the Rudin-Shapiro sequence, the ordinary  paperfolding sequence, the period-doubling sequence, and the Fibonacci sequence. Finally,  also show that the function counting the total number of palindromic factors in the prefix of length $n$ of a $k$-automatic sequence is not $k$-synchronized.
['Luke Schaeffer', 'Jeffrey Shallit']
I am writing a research paper on a idea Closed, Palindromic, Rich, Privileged, Trapezoidal, and Balanced Words in Automatic Sequences
[{"from": "human", "value": "I am writing a research paper on a idea Closed, Palindromic, Rich, Privileged, Trapezoidal, and Balanced Words in Automatic Sequences"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  prove that the property of being closed (resp., palindromic, rich, privileged trapezoidal, balanced) is expressible in first-order logic for automatic (and some related) sequences. It therefore follows that the characteristic function of those $n$ for which an automatic sequence $\\bf x$ has a closed (resp., palindromic, privileged, rich, trapezoidal, balanced) factor of length\u00a0 $n$ is itself automatic. For privileged words this requires a new characterization of the privileged property.  compute the corresponding characteristic functions for various famous sequences, such as the Thue-Morse sequence, the Rudin-Shapiro sequence, the ordinary\u00a0 paperfolding sequence, the period-doubling sequence, and the Fibonacci sequence. Finally,  also show that the function counting the total number of palindromic factors in the prefix of length $n$ of a $k$-automatic sequence is not $k$-synchronized."}]