2.0 KiB
2.0 KiB
| id | title | challengeType | forumTopicId | dashedName |
|---|---|---|---|---|
| 5900f4531000cf542c50ff65 | Problem 230: Fibonacci Words | 5 | 301874 | problem-230-fibonacci-words |
--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`.
```js
assert.strictEqual(fibonacciWords(), 850481152593119200);
```
# --seed--
## --seed-contents--
```js
function fibonacciWords() {
return true;
}
fibonacciWords();
```
# --solutions--
```js
// solution required
```