  We study critical percolation on a regular planar lattice. Let $E_G(n)$ be
the expected number of open clusters intersecting or hitting the line segment
$[0,n]$. (For the subscript $G$ we either take $\mathbb{H}$, when we restrict
to the upper halfplane, or $\mathbb{C}$, when we consider the full lattice).
Cardy (2001) (see also Yu, Saleur and Haas (2008)) derived heuristically that
$E_{\mathbb{H}}(n) = An + \frac{\sqrt{3}}{4\pi}\log(n) + o(\log(n))$, where $A$
is some constant. Recently Kov\'{a}cs, Igl\'{o}i and Cardy (2012) derived
heuristically (as a special case of a more general formula) that a similar
result holds for $E_{\mathbb{C}}(n)$ with the constant $\frac{\sqrt{3}}{4\pi}$
replaced by $\frac{5\sqrt{3}}{32\pi}$. In this paper we give, for site
percolation on the triangular lattice, a rigorous proof for the formula of
$E_{\mathbb{H}}(n)$ above, and a rigorous upper bound for the prefactor of the
logarithm in the formula of $E_{\mathbb{C}}(n)$.
