Pseudoknot RNA Structures with Arc-Length $\ge 3$

In this paper we study $k$-noncrossing RNA structures with arc-length $\ge 3$,i\mathrm{.}e\mathrm{.}RNAmoleculesinwhichforany$i$,thenucleotideslabeled$i$and$i+j$($j=1,2$)cannotformabondandinwhichthereareatmost$k-1$mutuallycrossingarcs\mathrm{.}Let${\sf S}{k,3}(n)$ denote their number. Based on a novel functional equation for the generating function $\sum{n\ge 0}{\sf S}{k,3}(n)z^n$,wederiveforarbitrary$k\ge 3$ exponential growth factors and for $k=3$ the subexponential factor. Our main result is the derivation of the formula ${\sf S}{3,3}(n) \sim \frac{6.11170\cdot 4!}{n(n-1)…(n-4)} 4.54920^n$.