# Problem 440: GCD and Tiling
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(ca), T(cb)) 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.