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.
最大公约数与摆方格
我们想要用1 × 2的空白方格或是1 × 1写有一个十进制数字的方格（如下所示）铺满长为n高为1的版面：
例如，下面是铺满长为n = 8的版面的一些方式：
记铺满长为n的版面的方式数为T(n)。
例如，T(1) = 10，T(2) = 101。
对于1 ≤ a, b, c ≤ L，记S(L)是三重求和∑a,b,c gcd(T(ca), T(cb)) 。
例如：
S(2) = 10444
S(3) = 1292115238446807016106539989
S(4) mod 987 898 789 = 670616280.
求 S(2000) mod 987 898 789。