We consider the sequence ${\left\{W_n\right\}}_{n=0}^{\infty }$, satisfying where u and v are positive integers with $gcd(u,v)=1$. The values of $W_n=W_n(u,v)$ depend on the values of u and v. If $u=v=1$, $F_n=W_n(1,1)$ is the n-th Fibonacci number [1]. If $u=1$ and $v=2$, $J_n=W_n(1,2)$ is the n-th Jacobsthal number [2,3]. If $u=2$ and $v=1$, $P_n=W_n(2,1)$ is the n-th Pell number [4]. However, for simplicity, if we do not specify the values of u or v, we will simply write $W_n$ for $W_n(u,v)$.

This type of number sequence has been well known to many people by Horadam’s series of studies ([5,6,7,8,9]) in the 1960s. Because of this fact, this sequence is sometimes called the Horadam sequence. Horadam himself used the recurrence relation $W_n=u{W}_{n-1}-v{W}_{n-2}$. But recently more people (see, e.g., [10,11]) have used the recurrence relation $W_n=u{W}_{n-1}+v{W}_{n-2}$ and such works are still due to Horadam. In general, the initial values are arbitrary, but because of some simplifications, we set $W_0=0$ and $W_1=1$. According to [6], this sequence has long exercised interest, as seen in, for instance, Bessel-Hagen [12], Lucas [13], and Tagiuri [14], and, for historical details, Dickson [15]. However, it is deplorable that quite a few papers are publishing results that have already been obtained by these authors as new results, either because they are unaware of their or the following important results, or even if they are ignoring them.

Given the set of positive integers $A:=\{a_1,a_2,\dots ,a_{\kappa }\}$ ($\kappa \ge 2$), for a nonnegative integer p, let $S_p$ be the set of integers whose nonnegative integral linear combinations of given positive integers $a_1,a_2,\dots ,a_{\kappa }$ are expressed in more than p ways. For a set of nonnegative integers ${\mathbb{N}}_0$, the set ${\mathbb{N}}_0\setminus S_p$ is finite if and only if $gcd(a_1,a_2,\dots ,a_{\kappa })=1$. Then there exists the largest integer $g_p\left(A\right):=g\left(S_p\right)$ in ${\mathbb{N}}_0\setminus S_p$, which is called the p-Frobenius number. The cardinality of ${\mathbb{N}}_0\setminus S_p$ is called the p-genus and is denoted by $n_p\left(A\right):=n\left(S_p\right)$. The sum of the elements in ${\mathbb{N}}_0\setminus S_p$ is called the p-Sylvester sum and is denoted by $s_p\left(A\right):=s\left(S_p\right)$. This kind of concept is a generalization of the famous Diophantine problem of Frobenius since $p=0$ is the case when the original Frobenius number $g\left(A\right)=g_0\left(A\right)$, the genus $n\left(A\right)=n_0\left(A\right)$ and the Sylvester sum $s\left(A\right)=s_0\left(A\right)$ are recovered. We can call $S_p$ the p-numerical semigroup. Strictly speaking, when $p\ge 1$, $S_p$ does not include 0 since the integer 0 has only one representation, so it satisfies simply additivity, and the set $S_p\cup \left\{0\right\}$ becomes a numerical semigroup. For numerical semigroups, we refer to [16,17,18]. For the p-numerical semigroup, we refer to [19].

We are interested in finding any closed or explicit form of the p-Frobenius number, which is even more difficult when $p>0$. For three or more variables, no concrete example had been found. Most recently, we have finally succeeded in giving the p-Frobenius number as closed-form expressions for the triangular number triplet ([20]), for repunits ([21,22]).

In this paper, we study the p-numerical semigroups of the triple $(W_i,W_{i+2},W_{i+k})$ for integers $i,k(\ge 3)$. We give explicit closed formulas of the p-Frobenius numbers and p-genus of this triple. Note that the special cases for Fibonacci [1], Pell [4], and Jacobsthal triples [2,3] have already been studied.

We introduce the Apéry set (see [23]) below in order to obtain the formulas for $g_p\left(A\right)$, $n_p\left(A\right)$, and $s_p\left(A\right)$ technically. Without loss of generality, we assume that $a_1=min\left(A\right)$.

It follows that for each p,

Even though it is hard to find any explicit form of $g_p\left(A\right)$ as well as $n_p\left(A\right)$ and $s_p\left(A\right)$$k\ge 3$, by using convenient formulas established in [24,25], we can obtain such values for some special sequences $(a_1,a_2,\dots ,a_{\kappa })$ after finding any regular structure of $m_j^{\left(p\right)}$. One convenient formula is on the power sum by using Bernoulli numbers $B_n$ defined by the generating function and another convenient formula is on the weighted power sum ([26,27]) by using Eulerian numbers $〈{\displaystyle \genfrac{}{}{0pt}{}{n}{m}}〉$ appearing in the generating function with $0^0=1$ and $〈{\displaystyle \genfrac{}{}{0pt}{}{0}{0}}〉=1$. Here, $\mu$ is a nonnegative integer and $\lambda \ne 1$. From these convenient formulas, many useful expressions are yielded as special cases. Some useful ones are given as follows. The formulas (3) and (4) are entailed from $s_{\lambda ,p}^{\left(0\right)}\left(A\right)$ and $s_{\lambda ,p}^{\left(1\right)}\left(A\right)$, respectively.

We use the following properties repeatedly. The proof is trivial and omitted.

First of all, if i is odd and $3\le i\le k-1$ then by (1) and (7), Hence, $g_0(W_i,W_{i+2},W_{i+k})=g_0(W_i,W_{i+2})$. Therefore, from now on, we consider the case only when i is even and k is odd, or when i is odd, with $i\ge k\ge 3$.

It is important to see that the elements of ${\mathrm{Ap}}_p\left(A\right)$ are determined from those of ${\mathrm{Ap}}_{p-1}\left(A\right)$.