A Recursive Block Pillar Structure in the Kolakoski Sequence K(1,3)
William Cook
TL;DR
This work reveals a recursive block–pillar construction for the Kolakoski sequence $K(1,3)$, showing that $B_{n+1}=B_n P_n B_n$ and $P_{n+1}=\mathcal{G}(R(P_n),3)$ yield prefixes $B_n$ that converge to $K(1,3)$ with the alternative relation $B_{n+1}=\mathcal{G}(R(B_n),1)$, linking self-encoding directly to the Kolakoski rule. Through exact recurrences for lengths, counts, and densities, the authors derive an exponential growth rate governed by the Pisot number $\alpha$, the real root of $x^3-2x^2-1=0$, with $\alpha\approx 2.20557$ and symbol frequency $d=(3-\alpha)/2\approx 0.397215$. A fundamental identity $m_n = L_n-2c_n$ ties pillar length to block composition, and a density analysis shows that if densities converge, the block and pillar densities coincide ($d=\delta$). The results connect the combinatorial recursion to substitution dynamics and model-set theory, offering a constructive viewpoint on the regularity of $K(1,3)$ and highlighting differences from the more mysterious $K(1,2)$.
Abstract
The Kolakoski sequence K(1,3) over {1, 3} is known to be structured, unlike K(1,2), with symbol frequency d approx. 0.397 linked to the Pisot number alpha (real root of x^3 - 2x^2 - 1 = 0). We reveal an explicit nested recursion defining block sequences B(n) and pillar sequences P(n) via B(n+1) = B(n) P(n) B(n) and P(n+1) = G(R(P(n)), 3), where G generates runs from vector R(P(n)). We prove B(n) are prefixes of K(1,3) converging to it, and B(n+1) = G(R(B(n)), 1), directly reflecting the Kolakoski self-encoding property. We derive recurrences for lengths |B(n)|, |P(n)| and symbol counts, confirming growth governed by alpha (limit |B(n+1)|/|B(n)| = alpha as n -> infinity). If block/pillar densities converge, they must equal d. This constructive framework provides an alternative perspective on K(1,3)'s regularity, consistent with known results from substitution dynamics.