# Problem 319: Bounded Sequences

![graphic](img319.gif)

Let x1, x2,..., xn be a sequence of length n such that: x1 = 2 for all 1
< i ≤ n : xi-1 < xi for all i and j with 1 ≤ i, j ≤ n : (xi) j
< (xj + 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(1010) and give your answer modulo 109.