--description--

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

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

Example:

Let $A = 1\,415\,926\,535$, $B = 8\,979\,323\,846$. We wish to find $D_{A,B}(35)$, say.

The first few terms of $F_{A,B}$ are:

$$\begin{align} & 1\,415\,926\,535 \\ & 8\,979\,323\,846 \\ & 14\,159\,265\,358\,979\,323\,846 \\ & 897\,932\,384\,614\,159\,265\,358\,979\,323\,846 \\ & 14\,159\,265\,358\,979\,323\,846\,897\,932\,384\,614\,15\color{red}{9}\,265\,358\,979\,323\,846 \end{align}$$

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

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

$$\begin{align} & 14\,159\,265\,358\,979\,323\,846\,264\,338\,327\,950\,288\,419\,716\,939\,937\,510 \\ & 58\,209\,749\,445\,923\,078\,164\,062\,862\,089\,986\,280\,348\,253\,421\,170\,679 \end{align}$$

and for $B$ the next hundred digits:

$$\begin{align} & 82\,148\,086\,513\,282\,306\,647\,093\,844\,609\,550\,582\,231\,725\,359\,408\,128 \\ & 48\,111\,745\,028\,410\,270\,193\,852\,110\,555\,964\,462\,294\,895\,493\,038\,196 \end{align}$$

Find $\sum_{n = 0, 1, \ldots, 17} {10}^n × D_{A,B}((127 + 19n) × 7^n)$.

--hints--

fibonacciWords() should return 850481152593119200.

assert.strictEqual(fibonacciWords(), 850481152593119200);

--seed--

--seed-contents--

function fibonacciWords() {

  return true;
}

fibonacciWords();

--solutions--

// solution required