Eccoci.

For any two strings of digits, $A$ and $B$, we define $F_{A, B}$ to be the sequence $(A,B,AB,BAB,ABBAB,\dots)$ in which each term is the concatenation of the previous two.

Further, we define $D_{A, B}(n)$ to be the $n$^th digit in the first term of $F_{A, B}$ that contains at least $n$ digits.

Example:

Let $A=1415926535$, $B=8979323846$. We wish to find $D_{A, B}(35)$, say.

The first few terms of $F_{A, B}$ are:
$1415926535$
$8979323846$
$14159265358979323846$
$897932384614159265358979323846$
$1415926535897932384689793238461415{\color{red}\mathbf 9}265358979323846$

Then $D_{A, B}(35)$ is the $35$^th digit in the fifth term, which is $9$.

Now we use for $A$ the first $100$ digits of $\pi$ behind the decimal point:

$14159265358979323846264338327950288419716939937510$
$58209749445923078164062862089986280348253421170679$

and for $B$ the next hundred digits:

$82148086513282306647093844609550582231725359408128$
$48111745028410270193852110555964462294895493038196$.

Find $\sum_{n = 0}^{17} 10^n \times D_{A,B}((127+19n) \times 7^n)$.