--description--

Let $x_1, x_2, \ldots, x_n$ be a sequence of length $n$ such that:

• $x_1 = 2$
• for all $1 < i ≤ n : x_{i - 1} < x_i$
• for all $i$ and $j$ with $1 ≤ i, j ≤ n : {(x_i)}^j < {(x_j + 1)}^i$

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 $10^9$.

--hints--

boundedSequences() should return 268457129.

assert.strictEqual(boundedSequences(), 268457129);

--seed--

--seed-contents--

function boundedSequences() {

  return true;
}

boundedSequences();

--solutions--

// solution required