Revised
Interesting question.
Here's a thought:
You can think of a ring, such as $\mathbb Z$, in terms of its monoid of affine endomorphisms
$x \rightarrow a x + b$. The action of this monoid, together with a choice for 0 and 1, give the structure of the ring. However, the monoid is not finitely generated, since the
multiplicative monoid of $\mathbb Z$ is the free abelian monoid on the the primes, times
the order 2 group generated by $-1$.
If you take a submonoid that uses only
one prime, it is quasi-isometric to a quotient of the hyperbolic plane by an action of $\mathbb Z$, which is multiplication by $p$ in the upper half-space model. To see this, place a dots labeled by integer $n$ at position $(n*p^k, p^k)$ in the upper half plane, for every pair of integers $(n,k)$, and connect them by horizontal line segments and by vertical line segments whenever points are in vertical alignment. The quotient of upper half plane by the hyperbolic isometry $(x,y) \rightarrow (p*x, p*y)$ has a copy of the Cayley graph for this monoid. This is also quasi-isometric to the 1-point union of two copies of the hyperbolic plane, one for negative integers, one for positive integes. It's a fun exercise,
using say $p = 2$. Start from 0, and recursively build the graph by connecting $n$ to $n+1$ by one color arrow, and $n$ to $2*n$ by another color arrow. If you arrange positive integers in a spiral, you can make a neat drawing of this graph (or the corresponding graph for a different prime.) The negative integers look just the same, but with the successor arrow reversed.
If you use several primes, the picture gets more complicated. In any case, one can take rescaled limits of these graphs, based at sequence of points, and get asymptotic cones for the monoid. The graph is not homogeneous, so there is not just one limit.
Another point of view is to take limits of $\mathbb Z$ without rescaling, but
with a $k$-tuple of constants $(n_1, \dots , n_k)$. The set of possible identities among polynomials in $k$ variables is compact, so there is a compact space of limit rings for $\mathbb Z$ with $k$ constants. Perhaps this is begging the question: the identitites that define the limits correspond to diophantine equations that have infinitely many solutions.
Rescaling may eliminate some of this complexity.
A homomorphism $\mathbb Z[x,y,\dots,z]/P$ to
$\mathbb Z$ gives a homomomorphism of the corresponding monoids, so an infinite sequence of these gives an action on some asymptotic cone for the affine monoid for $\mathbb Z$.
With the infinite set of primes, there are other plausible choices for how to define length; what's the best choice depends on whether and how one can prove anything of interest.
I echo Richard Stanley in his pessimism for there being a nice formula or method for giving you the solutions. On the other hand, most of the solutions will have a_i being 0 for most of the coefficients i. Here is another way to look at it, which might help you write a program for computer search for small n.
Your constraint summing to $n$ gives that $a_i + \epsilon_i = \frac{n}{i}$, where for most $i$, $0<\epsilon_i < 2$. Since you know most $a_i$ will be zero, $s$ will be not far from a sum of terms of the form $\frac{n(i-1)}{2}$,
so many solutions will be close to partitioning an integer near $\frac{2s}{n}$ into distinct parts (so e.g. a partition of 20 into {1,4,4,5,6} is not allowed). This roughening of the problem can be easily programmed without expanding the search space too much.
If you still hope for more, you might try deriving some recurrences which might help in
a partial characterization of solutions. It is my feeling that a quick program can be developed that will give you the solution set for any feasible s even for n as large as 50.
Gerhard "Email Me About System Design" Paseman, 2011.07.11
Best Answer
The number of solutions of $a_1x_1+\dots+a_mx_m=n$ in non-negative integers $x_1,\dots,x_m$, call it $d(n;a_1,\dots,a_m)$, is called the $\it denumerant$. This goes back to Sylvester, On the partition of numbers, Quart J Pure Appl Math 1 (1857) 141-152. Much is known. For example, Schur proves that if $\gcd(a_1,\dots,a_m)=1$ and $P_m=\prod a_i$ then $d(n;a_1,\dots,a_m)$ is asymptotic to $P_m^{-1}n^{m-1}/(m-1)!$ as $n\to\infty$. (The reference is Zur additiven zahlentheorie, Sitzungsberichte Preussiche Akad Wiss Phys Math Kl (1926) 488-495.)
This and more is in Chapter 4 of J L Ramirez Alfonsin, The Diophantine Frobenius Problem, published by Oxford.