An asymptotic study of growing altitude elementary paths of the cubic lattice Z³

Lets a_n be the number of growing altitude elementary paths of length n of the cubic lattice Z³. By numeric simulation shows that the quotient a_n+1/a_n tends rapidly to a constant. Leads to the decision that the sequence (a_n)_n has an asymptotically geometric behaviour. Confirms the intuition and shows that two positive constants α and λ exist, such that α_n = αλⁿ(1 + ε_n) where (ε_n)_n is a sequence tending to 0 as n tends to infinity with the estimation |ε_n| ≤ Cγⁿ where C > 0 and 0 < γ < 1. Explains the rapid convergence of a_n+1/a_n. Determines the constants α and λ and elaborates on a numeric method for their calculus.