Bounded Sequences

Let $x_1, x_2, \dots, x_n$ be a sequence of length $n$ such that:
$x_1 = 2$
for all $1 \lt i \le n$: $x_{i – 1} \lt x_i$
for all $i$ and $j$ with $1 \le i, j \le n$: $(x_i)^j \lt (x_j + 1)^i$.

There are only five such sequences of length $2$, namely:
$\{2,4\}$, $\{2,5\}$, $\{2,6\}$, $\{2,7\}$ and $\{2,8\}$.
There are $293$ such sequences of length $5$; three examples are given below:
$\{2,5,11,25,55\}$, $\{2,6,14,36,88\}$, $\{2,8,22,64,181\}$.

Let $t(n)$ denote the number of such sequences of length $n$.
You are given that $t(10) = 86195$ and $t(20) = 5227991891$.

Find $t(10^{10})$ and give your answer modulo $10^9$.

It appears that the problem involves a significant amount of computational mathematics and probably needs dynamic programming.

Let the sequence be $x_1, x_2, \dots, x_n$ and let $x_i = a$. Then the condition $(x_i)^j < (x_j + 1)^i$ simplifies to $a^j < (a+1)^i$ for all $i \ge j$. We can turn this into a programming task. We create a 2D dp array where dp[i][j] stands for the number of sequences length i which its last element is j. So, $dp[i][j]$ depends on all $dp[i-1][k]$ such that $k^i < (k+1)^{i-1}$. Here is a python script that may help illustrate this to achieve the solution: ```python MOD = 10 ** 9 maxn = 10 ** 10 maxm = 200 # We only need to check up to 200 because k^i rapidly becomes greater than (k+1)^(i-1) as k increases a = [[0]*maxm for _ in range(maxm)] dp = [[0]*maxm for _ in range(maxn+1)] for i in range(1, maxm): j = 1 while True: if pow(i, j) < pow(i+1, j-1): a[i][j] = True j += 1 else: break dp[0][0] = 1 # Initialization for i in range(1, maxn+1): sum_ = 0 for j in range(1, maxm): if a[j][i]: sum_ = (sum_ + dp[i-1][j-1]) % MOD dp[i][j] = sum_ print(dp[maxn][maxm-1]) ``` The code basically checks for every $i$ and every $j$, whether $j^i$ is smaller than $(j+1)^{i-1}$, which is equivalent to our initial condition. If it is, then it computes and adds it to the total combinations we can get for that $i$, keeping track of the sequences counted so far. Please note that this problem is a higher level combinatorial and programming problem and the given solution is Python-based, hence requires programming knowledge in Python.

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