Incidence of Double Letters in Fibonacci String

Theorem
Let $S_n$ denote the $n$th Fibonacci string.

Then:
 * $(1):\quad$ There are no instances of $2$ $\text a$'s together
 * $(2):\quad$ There are no instances of $3$ $\text b$'s together

in $S_n$.

Proof
The proof proceeds by strong induction.

For all $n \in \Z_{\ge 1}$, let $P \left({n}\right)$ be the proposition:
 * There are no instances of $2$ $\text a$'s or $3$ $\text b$'s together in $S_n$.

$P \left({1}\right)$ is the case:
 * $S_1 = \text a$

Basis for the Induction
$P \left({2}\right)$ is the case:
 * $S_2 = \text b$

$P \left({3}\right)$ is the case:
 * $S_3 = \text {ba}$

This is the basis for the induction.

Induction Hypothesis
Now it needs to be shown that, if $P \left({j}\right)$ is true, for all $j$ such that $2 \le j \le k$, then it logically follows that $P \left({k + 1}\right)$ is true.

This is the induction hypothesis:
 * There are no instances of $2$ $\text a$'s or $3$ $\text b$'s together in $S_k$.

and:
 * There are no instances of $2$ $\text a$'s or $3$ $\text b$'s together in $S_{k - 1}$.

from which it is to be shown that:
 * There are no instances of $2$ $\text a$'s or $3$ $\text b$'s together in $S_{k + 1}$.

Induction Step
This is the induction step:

By definition of Fibonacci string:
 * $S_{k + 1} = S_k S_{k - 1}$

concatenated.

By the induction hypothesis, neither $S_k$ nor $S_{k - 1}$ have any double $\text a$'s or triple $\text b$'s.

The only way for $S_{k + 1}$ to have a double $\text a$ is for $S_n$ to end with $\text a$ and for $S_{k - 1}$ to begin with $\text a$.

But from Fibonacci String Begins with ba, for all $k > 2$, $S_{k - 1}$ begins with $\text b$.

When $k = 2$, $S_{k - 1}$ does begin with $\text a$, but then $S_{k + 1} = \text {ba}$, which has no double $\text a$.

Similarly, the only way for $S_{k + 1}$ to have a triple $\text b$, either $S_k$ must end with double $\text b$ or $S_{k - 1}$ must begin with double $\text b$.

From Fibonacci String Begins with ba, $S_{k - 1}$ does not begin with double $\text b$.

From Fibonacci String Ends with ab or ba, $S_k$ does not begin with double $\text b$.

So $P \left({k}\right) \implies P \left({k + 1}\right)$ and the result follows by the Second Principle of Mathematical Induction.

Therefore:
 * For all $n \in \Z_{>0}$, there are no instances of $2$ $\text a$'s or $3$ $\text b$'s together in $S_n$.