1.6 KiB
1.6 KiB
| id | title | challengeType | forumTopicId | dashedName |
|---|---|---|---|---|
| 5900f4d51000cf542c50ffe8 | Problem 361: Subsequence of Thue-Morse sequence | 5 | 302022 | problem-361-subsequence-of-thue-morse-sequence |
--description--
The Thue-Morse sequence \\{T_n\\} is a binary sequence satisfying:
T_0 = 0T_{2n} = T_nT_{2n + 1} = 1 - T_n
The first several terms of \\{T_n\\} are given as follows: 01101001\color{red}{10010}1101001011001101001\ldots.
We define \\{A_n\\} as the sorted sequence of integers such that the binary expression of each element appears as a subsequence in \\{T_n\\}. For example, the decimal number 18 is expressed as 10010 in binary. 10010 appears in \\{T_n\\} (T_8 to T_{12}), so 18 is an element of \\{A_n\\}. The decimal number 14 is expressed as 1110 in binary. 1110 never appears in \\{T_n\\}, so 14 is not an element of \\{A_n\\}.
The first several terms of A_n are given as follows:
\begin{array}{cr}
n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & \ldots \\\\
A_n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 9 & 10 & 11 & 12 & 13 & 18 & \ldots
\end{array}$$
We can also verify that $A_{100} = 3251$ and $A_{1000} = 80\\,852\\,364\\,498$.
Find the last 9 digits of $\displaystyle\sum_{k = 1}^{18} A_{{10}^k}$.
# --hints--
`subsequenceOfThueMorseSequence()` should return `178476944`.
```js
assert.strictEqual(subsequenceOfThueMorseSequence(), 178476944);
```
# --seed--
## --seed-contents--
```js
function subsequenceOfThueMorseSequence() {
return true;
}
subsequenceOfThueMorseSequence();
```
# --solutions--
```js
// solution required
```