# Find total number of non negative integral solutions $5a + 6b + 9c + 2d + e = n$ with constraints

We want to find the total number of non negative integral solutions with the additional constraint that $$a + b \geq c + d$$

The value of $$n = \mathcal{O}(10^6)$$ and $$a, b, c, d, e \geq 0$$

I could use generating functions but I don't know how to solve with keeping the constraint satisfied? One approach that I thought of was first finding the total number of solutions to the equation $$5a + 6b + 9c + 2d + e = n$$ and then subtracting the ones that don't follow the constraint i.e. $$a + b < c + d$$. This means that $$a + b + k = c + d$$ where $$k \geq 0$$ but I don't know how I can use the slack variable $$k$$ to create another equation which satisfies all the constraints.

• Is the value of $n$ specified ?? Otherwise by setting $c=d = 0$ and changing the values of $a$ and $b$ , we can get infinite solutions. Oct 20, 2019 at 12:09
• Yes. The value of $n = \mathcal{O(10^6)}$ and we have to find non-negative integral solutions which means $a, b, c, d, e \geq 0$. Oct 20, 2019 at 12:18
• If I am not mistaken, for a given $n$, there are $$a_n=\sum_{k=0}^{\left\lfloor\frac{ n}5\right\rfloor}\sum_{\substack{{m\equiv n-5k\pmod{7}}\\ {0\leq m\leq n-5k}}}\sum_{c=0}^{\left\lfloor\frac{n-5k-m}{14}\right\rfloor}\Biggl(\min\left\{m,\frac{n+2k-m}{7}-c\right\}+1\Biggr)$$ solutions. I am not sure how to make this simpler. For each $(k,m,c)$, we get $$(a,b,c,d,e)=\left(\frac{n+2k-m}{7}-c-b,b,c,\frac{n-5k-m}{7}-2c,m-b\right),$$ where $$0\leq b \leq \min\left\{\frac{n+2k-m}{7}-c,m\right\}.$$ On the other hand, for each $(a,b,c,d,e)$, $$(k,m,c)=(a+b-c-d,b+e,c).$$ Oct 20, 2019 at 13:33
• Is there a way to convert the equation to an equivalent equations satisfying the given constraints? Oct 20, 2019 at 14:05