A very simple heuristic would be to  

- represent they circle as a directed, connected cycle graph $G(V,E)$ with vertices $V\ =\ \lbrace v_i\ |\ 1\le i\le n\rbrace$ and directed edges $E=\lbrace e_{ij}=(v_i,v_{i+1})\ |\ 1\le i \lt n\rbrace \cup\lbrace (v_n,v_1)\rbrace$  
- map each vertex $v_i$ to $A_i$
- while $E\ne\emptyset$  
 - determine edge $e_{i,j}^*$ that minimizes the sum $A_i^*+A_j^*$ of adjacent-vertex weights
 - record  $A_i^*+A_j^*$
 - assuming $(v_j^*,v_k)\in E$
     - $A_i^* := A_i^*+A_j^*$, 
     - $E = E\cup (v_i^*,v_k)$,
    - $V=V\setminus v_j^*$

That algorithm amounts to finding the pair of adjacent numbers with minimal weight sum,  
replace one of the numbers by that sum and remove the other from the circle.  

The time complexity is $O(n^2)$, which surely isn't optimal; utilizing priority queues will certainly bring it down to $O(n\log(n))$  

I like the question, because it poses different challenges like e.g. a linear programming formulation or to determine the most appropriate graph theoretic algorithm.  

running the algorithm with the sequence provided by the PO in a comment yields
$9,\ 4,\ (2+3),\ 2,\ 9\ \mapsto\ 9,\ 4,\ (5+2),\ 9\ \mapsto\ 9,\ (4+7),\ 9\ \mapsto\ 9,\ (11+9)\ \mapsto\ (9+20)$, resp.  
$9,\ 4,\ 2,\ (3+2),\ 9\ \mapsto\ 9,\ (4+2),\ 5,\ 9\ \mapsto\ 9,\ (6+5),\ 9\ \mapsto\ 9,\ (11+9)\ \mapsto\ (9+20)$  
the numbers generated by additon are $5,7,11,20,29$, resp. $5,6,11,20,29$, demonstrating that the heuristic fails to generate the optimal sequence of additions