## Bounded Sequences

### Problem 319

Published on Saturday, 8th January 2011, 07:00 pm; Solved by 251; Difficulty rating: 90%
Let `x`_{1}, `x`_{2},..., `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`+ 1)_{j}^{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}.