  For all positive non-square integer multiplier k, there is an infinity of
multiples of triangular numbers which are also triangular numbers. With a
simple change of variables, these triangular numbers can be found using
solutions of Pell equations. With some conditions on parities of fundamental
solutions of the simple and generalized Pell equations, only odd solutions of
the generalized Pell equation are retained to provide many infinitely solutions
found on branches corresponding to each of the generalized fundamental
solutions. General algebraic expressions of fundamental solutions of the Pell
equations are found for some values of the multiplier k in function of the
closest natural square. Further, among the expressions of Pell equation
solutions, a set of recurrent relations is identical to those found previously
without the Pell equation solving method. It is found also that two constants
of the problem of multiples of triangular numbers are directly related to the
fundamental solutions of the simple Pell equation, which is an unexpected
result as it means that simple Pell equation fundamental solutions in all
generality are related to constants in recurrent relations of the problem of
finding triangular numbers multiple of other triangular numbers.
