  The Kolakoski sequence is the unique infinite sequence with values in
$\{1,2\}$ and first term $1$ which equals the sequence of run-lengths of
itself, we call this $K(1,2).$ We define $K(m,n)$ similarly. A well-known
conjecture is that the limiting density of $K(1,2)$ is one-half. We state a
natural generalization, the "generalized uniformness conjecture" (GUC). The GUC
seems intractable, but we prove a partial result. The GUC implies that members
of a certain family of directed graphs $G_{m,n,k}$ are all strongly connected.
We prove this unconditionally.
  For $d>0,$ let $cf(m,n,d)$ be the density of indices $i$ such that $K(m,
n)_i=K(m, n)_{i+d}.$ Essentially, $cf(m, n, d)$ is the autocorrelation function
of a stationary stochastic process with random variables
$\{X_t\}_{t\in\mathbb{Z}}$ whereby a sample of a finite window of this process
is formed by copying as many consecutive terms of $K(m,n)$ starting from a
"uniformly random index" $i\in\mathbb{Z}_+.$ Assuming the GUC, we prove that we
can compute $cf(m,n,d)$ exactly for quite large $d$ by constructing a periodic
sequence $S$ of period around $10^{8.5}$ such that for $d$ not too large, the
correlation frequency at distance $d$ in $K(m,n)$ equals that in $S.$ We
efficiently compute correlations in $S$ using polynomial multiplication via
FFT.
  We plot our estimates $cf(m,n,d)$ for several small values of $(m,n)$ and
$d\le10^5$ or $10^6$. We note many suggested patterns. For example, for the
three pairs $(m,n)\in\{(1,2),(2,3),(3,4)\},$ the function $cf(m,n,d)$ behaves
very differently as we restrict $d$ to the $m+n$ residue classes $\text{mod}$
$m+n.$ The plots of the three functions $cf(1,2,d),cf(2,3,d),$ and $cf(3,4,d)$
resemble waves which have common nodes. We consider this very unusual behavior
for an autocorrelation function. The pairs $(m,n)\in\{(1,4),(1,6),(2,5)\}$ show
wave-like patterns with much more noise.
