Problem 319
Bounded Sequences

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.

These problems are part of Project Euler and are licensed under CC BY-NC-SA 2.0 UK

http://projecteuler.net/problem=319