Many people, including Horadam, have studied the numbers $W_n$, satisfying the recurrence relation $W_n=u W_{n-1}+v W_{n-2}$ ($n\ge 2$) with $W_0=0$ and $W_1=1$. In this paper, we study the $p$-numerical semigroups of the triple $(W_i,W_{i+2},W_{i+k})$ for integers $i,k(\ge 3)$. 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.

In this paper, we give closed-form expressions of the $p$-Frobenius number for the triple of the generalized star numbers $a n(n-1)+1$ for an integer $a\ge 4$. When $a=6$, it is reduced to the famous star number. For the set of given positive integers $\{a_1,a_2,\dots,a_k\}$, the $p$-Frobenius number is the largest integer $N$ whose number of nonnegative integer representations $N=a_1 x_1+a_2 x_2+\dots+a_k x_k$ is at most $p$. When $p=0$, the $0$-Frobenius number is the classical Frobenius number, which is the central topic of the famous linear Diophantine problem of Frobenius.