To clarify, here are examples of binary sequences that contain three consecutive identical blocks of numbers:
$\color{red}{111}$
$101\color{red}{000}01$
$00\color{red}{100100100}011$
There are exactly $6$ binary sequences that do not contain two consecutive identical blocks of numbers:
$0$
$1$
$01$
$10$
$010$
$101$
My question is (again), how many binary sequences do not contain three consecutive identical blocks of numbers? I know there are $2$ such sequences of length $1$; $4$ such sequences of length $2$; $6$ such sequences of length $3$; $10$ such sequences of length $4$. But for greater lengths, I do not know how to count the number of such sequences. I presume that beyond a certain length, there are no such sequences.
(According to another question, which inspired this question, there are an infinite number of binary sequences that do not contain $six$ consecutive identical blocks of numbers. I also wonder how many binary sequences do not contain four, or five, consecutive identical blocks of numbers; but one question at a time.)
Best Answer
Shortly after posting my question, I found the answer: infinite, because the Thue-Morse sequence is cubefree and arbitrarily long.