Number of integral solutions for $a_1x_1+a_2x+_2+a_3x_3+\ldots+a_nx_n<k$

Question

Given $a = [a_1, a_2, a_3, a_4, \ldots, a_n]$ where $a_i$ is positive integer, I need to find the number of non negative integral solutions for the equation $$a_1x_1+a_2x+_2+a_3x_3+\ldots+a_nx_n<k$$ where $k>0$.

Answer 1

If all $a_i$ are positive (or more generally non-negative and $a_n$ is positive), then $x\mapsto a_1x+a_2x^2+\cdots +a_nx^n$ is a strictly increasing function. Then the problem amounts to finding the minimal positive integer $x$ for which this polynomial expression is $\ge k$. A simple upper bound, good enough, e.g., for a binary search, is $\lceil\sqrt[n]{\frac k{a_n}}\rceil$.