  For a nonnegative integer $p$, the $p$-numerical semigroup $S_p$ is defined
as the set of integers whose nonnegative integral linear combinations of given
positive integers $a_1,a_2,\dots,a_\kappa$ with
$\gcd(a_1,a_2,\dots,a_\kappa)=1$ are expressed in more than $p$ ways. When
$p=0$, $S=S_0$ is the original numerical semigroup. The largest element and the
cardinality of $\mathbb N_0\backslash S_p$ are called the $p$-Frobenius number
and the $p$-genus, respectively. Their explicit formulas are known for
$\kappa=2$, but those for $\kappa\ge 3$ have been found only in some special
cases. For some known cases, such as the Fibonacci and the Jacobsthal triplets,
similar techniques could be applied and explicit formulas such as the
$p$-Frobenius number could be found.
  In this paper, we give explicit formulas for the $p$-Frobenius number and the
$p$-genus of Pell numerical semigroups
$\bigl(P_i(u),P_{i+2}(u),P_{i+k}(u)\bigr)$. Here, for a given positive integer
$u$, Pell-type numbers $P_n(u)$ satisfy the recurrence relation $P_n(u)=u
P_{n-1}(u)+P_{n-2}(u)$ ($n\ge 2$) with $P_0(u)=0$ and $P_1(u)=1$. The
$p$-Ap\'ery set is used to find the formulas, but it shows a different pattern
from those in the known results, and some case by case discussions are
necessary.
