## GCD and Tiling

### Problem 440

Published on Sunday, 13th October 2013, 07:00 am; Solved by 188; Difficulty rating: 60%We want to tile a board of length `n` and height 1 completely, with either 1 × 2 blocks or 1 × 1 blocks with a single decimal digit on top:

For example, here are some of the ways to tile a board of length `n` = 8:

Let T(`n`) be the number of ways to tile a board of length `n` as described above.

For example, T(1) = 10 and T(2) = 101.

Let S(`L`) be the triple sum ∑_{a,b,c} gcd(T(`c`^{a}), T(`c`^{b})) for 1 ≤ `a`, `b`, `c` ≤ `L`.

For example:

S(2) = 10444

S(3) = 1292115238446807016106539989

S(4) mod 987 898 789 = 670616280.

Find S(2000) mod 987 898 789.