Let $S_n$ be a sequence of integers with $n\geq 0$ such that $S_0=S_1=S_2=0$ (also assume $S_n$ to not be completely zero) and $$ S_{n-1}+S_{n+1}\geq 2S_n $$ i.e. $S_n$ is convex. Now consider the following quantity $$ \Delta(a,b,c)=S_{a+b+c} - (S_{a+b}+S_{a+c}+S_{b+c})+S_a+S_b+S_c $$ what additional conditions do I need to impose on $S_n$ to gurantee that $\Delta(a,b,c)\geq 0$ for all triples? Note that if the positivity condition holds, with $S_1=S_2=0$, then $\Delta(1,1,n-1)=S_{n+1}-2S_n+S_{n-1}\geq 0$. So convexity is a consequence of positivity.
Addendum: To give a little context, what interests me are only those $S_n$ that have a "periodicity" property. Defining $\ell_n = S_n-S_{n-1}$ for $n\geq 1$, then there exist a pair of positive integers $v$ and $d$ (with $d\geq 2$), such that $$ \ell_{n+v} = \ell_n +d $$ The $S_n$ sequence is called "pattern of zeros" in the language of quantum Hall effect. In the physics literature, both periodicity and positivity of $\Delta$ are predictions of the theory and convexity of $S_n$ is only a mathematical consequence. I'm curious if there is a way to flip the cause and effect by adding extra conditions to periodicity and convexity.
Addendum 2: There is yet another physical property which might ease the problem. For $k=0,\cdots, v$, we have $$ S_{v-k}-S_k = S_v - k \left(\frac{2S_v}{v}\right) $$ and furthermore $2S_v/v$ is an integer. In other words, $S_n$ also has a "reflective" property, in some sense.