Journal of work on two-dimensional recurrences

1. Frieze patterns

The "frieze pattern" recurrence is given by

  a
b   c    ad = bc + 1
  d

or (rotated)

a  b
         ad = bc + 1
c  d

I decided to look at the "square" initial conditions: x_ij is given for i = 0 or j = 0. I wrote Mathematica code (nb, txt) to solve this recurrence and represent the monomials in an easy-to-read format; here is some output. From this output it was easy to conjecture a general form for the solution:

       x_i0 * x_0j     i       1           j       1        
x_ij = ----------- + (sum -------------) (sum -------------).
          x_00        k=1 x_k0 x_(k-1)0   k=1 x_0k x_0(k-1)

Reading the antifrieze notes revealed the combinatorial origin of this formula: the monomials correspond to paths through the graph

+---+---+   +---+
|x00|x01|...|x0j|
+---+---+   +---+
|x10|
+---+
  .
  .
+---+
|xi0|
+---+

from the lower-left vertex to the upper-right vertex; the exponent of each variable is related to the number of edges in the path which touch it.

The combinatorial interpretation of these frieze patterns seems to be fairly well understood, so I decided to look at other two-dimensional recurrences.

2. Number walls

To be continued...

3. A modified number-wall recurrence

To be continued...