## Titanic sets

### Problem 415

Published on Sunday, 17th February 2013, 10:00 am; Solved by 177; Difficulty rating: 100%A set of lattice points S is called a *titanic set* if there exists a line passing through exactly two points in S.

An example of a titanic set is S = {(0, 0), (0, 1), (0, 2), (1, 1), (2, 0), (1, 0)}, where the line passing through (0, 1) and (2, 0) does not pass through any other point in S.

On the other hand, the set {(0, 0), (1, 1), (2, 2), (4, 4)} is not a titanic set since the line passing through any two points in the set also passes through the other two.

For any positive integer `N`, let `T`(`N`) be the number of titanic sets S whose every point (`x`, `y`) satisfies 0 ≤ `x`, `y` ≤ `N`.
It can be verified that `T`(1) = 11, `T`(2) = 494, `T`(4) = 33554178, `T`(111) mod 10^{8} = 13500401 and `T`(10^{5}) mod 10^{8} = 63259062.

Find `T`(10^{11}) mod 10^{8}.