  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.
