Let H(n) be the number of simply connected subsets with n hexagons in an infinite hexagonal network. Also let P(n) be the number of paths starting from a fixed vertex (that do not connect itself) with lentgh n in this hexagonal network.
a) Prove that the limits \alpha: \equal{}\lim_{n\rightarrow\infty}H(n)^{\frac1n}, \beta: \equal{}\lim_{n\rightarrow\infty}P(n)^{\frac1n}exist.
b) Prove the following inequalities:
2≤β≤2
α≤12.5
α≥3.5
α≤β4 limitinequalitiescombinatorics proposedcombinatorics