The Strange World of the Hausdorff Metric Geometry

# XVI. Polygonal Chain Configurations

In 2006, Lisa Morales, Dan Schultheis, and I explored a new family of finite configurations, the polygonal chains. A polygonal chain #(PCm,lk) is constructed by chaining together k copies of m polygonal configurations with string configurations of length l. Two examples are shown below.

Figure 11

Top: PC2,23; Bottom: : PC3,22

If we fix two of the parameters k, m, and l and let the other vary, we obtain an infinite sequence of integers #(PCm,lk).

For k = 1 we have

#(PCm,l) = #(Pm) = L2m and #(PCm,l+S1) = #(Pm+S1) = F2m+2.

Using a lot of induction and some techniques from linear algebra in [22], we found that for m ≥ 2, k ≥ 2, and l ≥ 1,

#(PCm,1k) = Rm,l λm,lk-1 + Rm,l μm,lk-1,

where λ and μ (λ > μ) are the eigenvalues of the matrix M = [mp,q], with

m1,1 = L2mFl-2 + F2m+2Fl-1,  m1,2 = L2mFl-1 + F2m+2Fl

m2,1 = F2m+2Fl-2 + Fl-1(Fm+2)2,  m2,2 = F2m+2Fl-1 + Fl(Fm+2)2

and

, and .

In these formulas we let F-1 = 1. The table below shows a few entries in these polygonal chain sequences.

 i 1 2 3 4 5 P2i(S1) 7 113 1815 29153 468263 Pi2(S1) 113 765 5234 35865 P22(Si) 113 176 289 465 754 P2i(S2) 7 176 4393 109649 2736832 Pi3(S2) 4393 80361 1425131 25671393 P33(Si) 32733 80361 215658 559305 1469565

The polygonal chains provide an infinite three-parameter family of previously unknown integer sequences, some of which can now be found in the On-Line Encyclopedia of Integer Sequences.