# XVII. Polygonal Chain Sequences, Asymptotic Behavior

The numbers in the polygonal chain sequences appear to increase quite rapidly. The successive quotients as shown in the following table indicate that the behavior of these sequences is asymptotically exponential. For the sake of convenience, let Xi,m,l = #(PCm, li+1) / #(PCm, li), Yk,i,l = #(PCi+1, lk) / #(PCi, lk), and Zk,m,i = #(PCm, i+1k) / #( PCm, ik).

 i 6 7 8 9 10 Xi,2,1 16.0623 16.0623 16.0623 16.0623 16.0623 Y2,i,1 6.85406 6.8541 6.8541 6.8541 6.8541 Z2,2,i 1.61854 1.61784 1.61811 1.61801 1.61804 Xi,2,2 24.9599 24.9599 24.9599 24.9599 24.9599 Y3,i,2 17.9547 17.9402 17.9458 17.9437 17.9445 Z3,3,i 2.61941 2.61751 2.61823 2.61796 2.61806

In fact, these sequences are asymptotically exponential, and the limits of the ratios of consecutive terms are given in [21] as follows:

·         Xi,m,l → λm,l

·          Yk,i,l → φ2k

·         Zk,m,i → φk-1

where φ is the golden ratio and λm,l is as defined on the previous page.

For example, compare the table entries to the values  λ2,1 = 8 + √65 » 16.06225775 and λ2,2 = (25 + 3√69)/2 » 24.95993579.