Let
(
U
n
)
n
∈
N
(U_n)_{n\in \mathbb {N}}
be a fixed linear recurrence sequence defined over the integers (with some technical restrictions). We prove that there exist effectively computable constants
B
B
and
N
0
N_0
such that for any
b
,
c
∈
Z
b,c\in \mathbb {Z}
with
b
>
B
b> B
the equation
U
n
−
b
m
=
c
U_n - b^m = c
has at most two distinct solutions
(
n
,
m
)
∈
N
2
(n,m)\in \mathbb {N}^2
with
n
≥
N
0
n\geq N_0
and
m
≥
1
m\geq 1
. Moreover, we apply our result to the special case of Tribonacci numbers given by
T
1
=
T
2
=
1
T_1= T_2=1
,
T
3
=
2
T_3=2
and
T
n
=
T
n
−
1
+
T
n
−
2
+
T
n
−
3
T_{n}=T_{n-1}+T_{n-2}+T_{n-3}
for
n
≥
4
n\geq 4
. By means of the LLL-algorithm and continued fraction reduction we are able to prove
N
0
=
2
N_0=2
and
B
=
e
438
B=e^{438}
. The corresponding reduction algorithm is implemented in Sage.