Formula for runs of a fixed part and length in compositions

Let $R(n,k,l)$ denote the number of runs of $k$ of length $l$ over all compositions of $n$, and let $a(1,m)$ be the corresponding two-toned tiling count, interpreted as zero when its second argument is negative. The run-count formula. For all $n,k,l$,

$$R(n,k,l)=a(1,n-kl)-2a(1,n-(l+1)k)+a(1,n-(l+2)k).$$

This gives an explicit relation between refined run counts in compositions and two-toned tiling numbers; the supplied context does not indicate that the formula is open or conjectural.