---
id: 5900f4d51000cf542c50ffe8
title: 'Problem 361: Subsequence of Thue-Morse sequence'
challengeType: 5
forumTopicId: 302022
dashedName: problem-361-subsequence-of-thue-morse-sequence
---

# --description--

The Thue-Morse sequence $\\{T_n\\}$ is a binary sequence satisfying:

- $T_0 = 0$
- $T_{2n} = T_n$
- $T_{2n + 1} = 1 - T_n$

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
```