  We define the generalized Golomb triangular recursion by g_{j,s,lambda}(n) =
g_{j,s,lambda}(n - s - g_{j,s,lambda}(n-j)) + \lambda j. For particular choices
of the initial conditions, we show that the solution of the recursion is a
non-slow monotone sequence for which we can provide a combinatorial
interpretation in terms of a weighted count of the leaves of a certain labeled
infinite tree. We discover that more than one such tree interpretation is
possible, leading to different choices of the initial conditions and
alternative solutions that are closely related. In the case lambda=1 the
initial conditions for these alternative tree interpretations coincide and we
derive explicit closed forms for the solution sequence and its frequency
function.
