We give a sufficient and necessary condition on the sequence
{
a
n
}
\{a_n\}
of integers that for any integer
l
≥
1
l\ge 1
, every integer can be represented in the form
ε
l
a
l
+
ε
l
+
1
a
l
+
1
+
⋯
+
ε
k
a
k
\varepsilon _l a_l+\varepsilon _{l+1} a_{l+1}+\cdots + \varepsilon _ka_k
, where
ε
i
∈
{
−
1
,
1
}
(
i
=
l
,
l
+
1
,
…
,
k
)
\varepsilon _i\in \{-1, 1\}\ (i=l,l+1,\ldots , k)
. This generalizes the known result on integral-valued polynomial values. Moreover, we show that such sequences exist with any growth rate. This answers two problems posed by Bleicher. We also pose several problems for further research.