  A Cullen number is a number of the form $m2^m+1$, where $m$ is a positive
integer. In 2004, Luca and St\u anic\u a proved, among other things, that the
largest Fibonacci number in the Cullen sequence is $F_4=3$. Actually, they
searched for generalized Cullen numbers among some binary recurrence sequences.
In this paper, we will work on higher order recurrence sequences. For a given
linear recurrence $(G_n)_n$, under weak assumptions, and a given polynomial
$T(x)\in \mathbb{Z}[x]$, we shall prove that if $G_n=mx^m+T(x)$, then \[
m\ll\log \log |x|\log^2(\log \log |x|)\ \mbox{and}\ n\ll\log |x|\log\log
|x|\log^2(\log \log |x|), \] where the implied constant depends only on
$(G_n)_n$ and $T(x)$.
