Let x_{1}, x_{2},..., x_{n} be a sequence of length n such that:
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}.
These problems are part of Project Euler and are licensed under CC BY-NC-SA 2.0 UK