calculating Markoff numbers with matrices
can be calculated with the aid of particular kinds of continued fractions
(see Frobenius for details).
However, analogous calculations can be achieved with matrix multiplication,
which provides a simpler illustration of the process.
So, although I like continued fractions, I will use matrices here.
First, define two matrices, a and b, as follows.
Below them are matrices A and B, formed by
squaring a and b (respectively).
Any Markoff number (except for 1 or 2) can be calculated
by taking a particular product of these matrices and examining the
upper-left element of the resulting product matrix. For example:
7561 is the upper-left element in the matrix
generated as the product bABAABAb. (This kind of notation for
the product of a string of matrices is unambiguous, as matrix
multiplication is associative.) The bABAABAb matrix is as follows:
Next to each Markoff number in the tree below is the form of the matrix
product (in blue) which generates it:
Each matrix product expression (bb, bABAb, etc.) is
palindromic. Each expression begins and ends with
matrix b; all other terms are A and B matrices. The initial
and final b matrices are shown in light blue above; the patterns
of A and B terms (in dark blue) are the interesting part. The
patterns of A and B terms exhibit the following properties:
The tree has a complementary symmetry about its center line: the pattern
of A and B terms for any number in the tree is the complement
of the pattern for the number in the corresponding location in the other
half of the tree. E.g, 7561 (ABAABA) and 37666 (BABBAB).
The Fibonacci numbers along the bottom edge of the tree have
patterns comprised solely of A terms
(A, AA, AAA, ...); each pattern has one more A term than its
predecessor. The Pell numbers on the other
side of the tree have complementary patterns comprised
solely of B terms (B, BB, BBB, ...).
Two conjectures (which I've since been told are both provable):
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