Sylvester sums on the Frobenius set in arithmetic progression

Let $a_1,a_2,\dots,a_k$ be positive integers with $\gcd(a_1,a_2,\dots,a_k)=1$. The concept of the weighted sum $\sum_{n\in{\rm NR}}\lambda^{n}$ is introduced in \cite{KZ0,KZ}, where ${\rm NR}={\rm NR}(a_1,a_2,\dots,a_k)$ denotes the set of positive integers nonrepresentable in terms of $a_1,a_2,\dots,a_k$. When $\lambda=1$, such a sum is often called Sylvester sum. The main purpose of this paper is to give explicit expressions of the Sylvester sum ($\lambda=1$) and the weighed sum ($\lambda\ne 1$), where $a_1,a_2,\dots,a_k$ forms arithmetic progressions. As applications, various other cases are also considered, including weighted sums, almost arithmetic sequences, arithmetic sequences with an additional term, and geometric-like sequences. Several examples illustrate and confirm our results.