We investigate here sums of triangular numbers $f(x):=\ssum{i}{} b_i T_{x_i}$
where $T_n$ is the $n$-th triangular number. We show that, fixing $b_i\geq 0$,
$f(x)$ represents every nonnegative integer if and only if it represents 1, 2,
4, 5, and 8, with the standard application to sums of odd squares $\ssum{i}{}
b_i (2x_i+1)^2$. Moreover, we show that no finite subset will suffice if "cross
terms" are included, in turn showing that there is no overarching finiteness
theorem which generalizes from positive definite quadratic forms to totally
positive quadratic polynomials.

The average value of log s(n)/n taken over the first N even integers is shown
to converge to a constant lambda when N tends to infinity; moreover, the value
of this constant is approximated and proven to be less than 0. Here s(n) sums
the divisors of n less than n. Thus the geometric mean of s(n)/n, the growth
factor of the function s, in the long run tends to be less than 1. This could
be interpreted as probabilistic evidence that aliquot sequences tend to remain
bounded.