The Markov numbers 
are the union of the solutions 
to the Markov equation
 |
(1)
|
and are related to Lagrange numbers 
by
 |
(2)
|
The first few solutions are 
, (1, 1, 2), (1, 2, 5), (1, 5, 13), (2, 5, 29),
.... All solutions can be generated from the first two of these since the equation
is a quadratic in each of the variables, so one integer solution leads to a second,
and it turns out that all solutions (other than the first two singular ones) have
distinct values of 
,
, and 
, and share two of their three values with three other solutions
(Guy 1994, p. 166). The Markov numbers are then given by 1, 2, 5, 13, 29, 34,
... (OEIS A002559).
The Markov numbers for triples 
in which one term is 5 are 1, 2, 13, 29, 194, 433, ...
(OEIS A030452), whose terms are given by the
recurrence relation
 |
(3)
|
with 
,
, 
, and 
.
The solutions can be arranged in an infinite tree with two smaller branches on each trunk. It is not known if two different regions can have the same label. Strangely, the regions adjacent to 1 have alternate Fibonacci numbers 1, 2, 5, 13, 34, ..., and the regions adjacent to 2 have alternate Pell numbers 1, 5, 29, 169, 985, ....
Let 
be the number of triples with 
, then
 |
(4)
|
where 
(Guy 1994, p. 166).
