  For a non-negative integer $p$, one of the generalized Frobenius numbers,
that is called the $p$-Frobenius number, is the largest integer that is
represented at most in $p$ ways as a linear combination with nonnegative
integer coefficients of a given set of positive integers whose greatest common
divisor is one. The famous so-called Frobenius number proposed by Frobenius is
reduced to the $0$-Frobenius number when $p=0$. The explicit formula for the
Frobenius number with two variables was found in the 19th century, but a
formula with more than two variables is very difficult to find, and closed
formulas of Frobenius numbers have been found only in special cases such as
geometric, Thabit, Mersenne, and so on. The case of $p>0$ was even more
difficult, and not a single formula was known. However, most recently, we have
finally succeeded in giving the $p$-Frobenius numbers as closed-form
expressions of the triangular number triplet , repunits, Fibonacci triplet and
Jacobsthal triplet.
  In this paper, we give closed-form expressions of the $p$-Frobenius number
for the finite sequence $\{a b^n-c\}_n$, where $a$, $b$ and $c$ are integers
with $a\ge 1$, $b\ge 2$ and $c\ne 0$. This sequence includes the cases for
geometric, Thabit and Mersenne as well as their variations.
