Kolakoski Sequence

Let $K(a,b)$ be the infinite sequence such that:

• The first element is $a$,
• It consists of $a$ and $b$, and
• Its run length sequence is equal to $K(a,b)$.

Here, for example, we assume that the run length sequence of $[1,1,1,2,3,4,4,5]$ is $[3,1,1,2,1]$.

We can verify that

• $K(2,3)=[2,2,3,3,2,2,2,3,3,3,2,\dots ]$, and
• $K(2,4)=[2,2,4,4,2,2,2,2,4,4,4,4,2,\dots ]$.

Let $C_{a,b}(N)$ be the number of $a$ in the first $N$ elements of $K(a,b)$.

For example, $C_{2,4}(4)=2$ and $C_{2,4}(5)$ = 3.

Let $S(M,N):=\sum _{a=2}^{M-1}\sum _{b=a+1}^M{C}_{a,b}(N)$.

You are given $S(5,100)=300$ and $S(10,{10}^6)=17856847$.

Find $S(100,{10}^{15})$.