AN ASYMPTOTIC STUDY OF GROWING ALTITUDE ELEMENTARY PATHS OF THE CUBICLATTICE Z(3)

Lets a(n) be the number of growing altitude elementary paths of length n of the cubic lattice Z(3). By numeric simulation shows that the quo tient a(n+1)/a(n) tends rapidly to a constant. Leads to the decision t hat the sequence (a(n))(n) has an asymptotically geometric behaviour. Confirms the intuition and shows that two positive constants alpha and lambda exist, such that alpha(n) = alpha lambda(n)(1 + epsilon(n)) wh ere (epsilon(n))(n) is a sequence tending to 0 as n tends to infinity with the estimation \ epsilon(n) \ less than or equal to C gamma(n) wh ere C > 0 and 0 < gamma < 1. Explains the rapid convergence of a(n+1)/ a(n). Determines the constants alpha and lambda and elaborates on a nu meric method for their calculus.