Exact densities of loops in O(1) dense loop model and of clusters in critical percolation on a cylinder

We obtain exact densities of contractible and non-contractible loops in the O(1) model on a strip square lattice rolled into an infinite cylinder finite even circumference $L$. They are also equal to critical percolation clusters forty five degree rotated cylinder, which do not or wrap around respectively. The results presented as explicit rational functions $L$ taking values for any Their asymptotic expansions large limit have irrational coefficients reproducing earlier leading orders. solution is based mapping six-vertex use technique Baxter's T-Q equation.