The first factorization discovered (and "proved") in James Sellers' paper

           Domino Tilings and Products of Fibonacci and Pell Numbers

published in        Journal of Integer Sequences, Vol. 5 (2002), Article 02.\
    1.2, is:

                       [[1, 2], [1, 1]], [[2, 5], [2, 1]]

      The second factorization discovered (and "proved") in that paper is:

                      [[1, 2], [3, -1]], [[1, 4], [5, -1]]

The third factorization discovered (and whose "proof" was left to the reader\
    , since it was too "complicated") in that paper is:

[[1, 1], [1, 1]], [

    [1, 1, 17, 51, 449, 1853, 12853, 61557], [1, 25, 11, -47, -11, 25, -1, -1]]

                          This took, 0.487, seconds .