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\begin{center} 
\vskip 1cm{\LARGE\bf  
Analytic Representations  of the $\it n$-anacci \\  
\vskip .1in 
Constants and Generalizations Thereof} 
 
\vskip 1cm 
\large 
Igor Szczyrba \\ 
School of Mathematical Sciences \\ 
University of Northern Colorado \\ 
Greeley, CO 80639 \\
USA \\
\href{mailto:igor.szczyrba@unco.edu}{\tt igor.szczyrba@unco.edu} \\ 
\ \\ 
Rafa\l\ Szczyrba\\ 
Funiosoft, LLC   \\ 
Silverthorne, CO 80498\\
USA \\
\href{mailto:rafals@funiosoft.com}{\tt rafals@funiosoft.com} \\ 
\ \\ 
Martin Burtscher \\ 
Department of Computer Science \\ 
Texas State University \\ 
San Marcos, TX 78666 \\
USA \\
\href{mailto:burtscher@txstate.edu}{\tt burtscher@txstate.edu} \\ 
\end{center} 
 
\vskip .2 in 
 
\begin{abstract} 
We study generalizations of the sequence of the $n$-anacci constants that are constructed from the ratio limits generated by linear recurrences of an arbitrary  order $n$ with equal integer weights $m$. We  derive the  analytic representation of the class $C^\infty$\! of  these ratio limits and  prove  that, for  a fixed $m$,  the ratio limits form  a strictly increasing  sequence converging  to $m\!+\!1$. We also show that the generalized $n$-anacci constants form a totally ordered set. 
\end{abstract}
  
\section{Introduction} 
 
We study properties of the ratio limits  $\Phi^{(n)}(m)$, $m,n\!\in\!\mathbb N$,  of the successive terms of the integer sequences $\big(F^{(n)}_k(m)\big)_{k=1}^\infty$ with the signatures $(m,\dots,m)$ 
generated  by the linear recurrences $F^{(n)}_k(m)$ of an arbitrary order $n$ with equal weights $m$, i.e.,   
\begin{equation}  
    F^{(n)}_k(m)\!\equiv\!m\big(F^{(n)}_{k-1}(m)\!+\!\cdots\!+\!F^{(n)}_{k-n}(m)\big),\,\,  n\!\leq\!k,\quad \textrm{and} \quad F^{(n)}_{k\!}(m)\!=\! a_k\!\in\! \mathbb{N},\,\, 0\!\le\! k \!&lt;\! n, \label{11} 
\end{equation} 
\begin{equation} 
 \Phi^{(n)}(m)\!\equiv\!\lim_{k\to\infty} F^{(n)}_{k+1}(m)/F^{(n)}_{k}(m),\,\,k &gt; k_0,  \label{12} 
\end{equation} 
where  $k_0$ is the largest index for which  $F^{(n)}_{k_0}(m)\!=\!0$. 
 
\medskip The linear recurrences \eqref{11}  generate, in particular, the following sequences:  
\begin{enumerate} 
\item The $n$-step Fibonacci sequences, with the signatures $(1,\dots,1)$, introduced in 1960 by Miles \cite{Miles}  as the $n$-generalized Fibonacci sequences, and further  investigated by Flores  \cite{Flores} and  Dubeau \cite{Dubeau1}, as well as more recently by Zhu and Grossman \cite  
{Zhu}. 
\item 
 The $n$-step Lucas sequences, with the signatures $(1,\dots,1)$, introduced in 1967  by Fiedler \cite{Fiedler}, and recently studied by Catalani \cite{Catalani}, Benjamin and Quin \cite{Benjamin}, as well as by Noe and Post \cite{Noe1}. 
\item The  2-step  sequences with the signatures $(m,m)$ that are a special case of  the Horadam sequences \cite{Horadam1,Horadam2}. 
\end{enumerate} 
 
\smallskip 
Sloane  \cite{Sloane} and Khovanova \cite{Khovanova} catalog numerous sequences with the signatures  $(1,\dots,1)$, $2\!\le\!n\!\le\!13$, and a large variety of  initial conditions; the Horadam sequences with the signatures $(m,m)$, $2\!\le\!m\!\le\!10$, and various initial conditions;  as well as several sequences with the signatures $(m,\dots,m)$, $n\!&gt;\!2$, generated by the linear recurrences \eqref{11}.   
 
Following the  terminology that refers to the elements of the sequence $\big(\Phi^{(n)}(1)\big)_{n=1}^\infty$   as the $n$-anacci constants \cite{Noe2},  we  call  the  elements of the set $\big\{\Phi^{(n)}(m)\,|\, m,n\!\in\!\mathbb{N}\big\}$  the $(m,n)$-anacci constants. 
 
In 1966, Ostrowski showed that   polynomials  
\begin{equation} 
  P^{(n)}(\lambda)\!\equiv\!\lambda^{n}\!-\!b_1\lambda^{n-1}\!-\!\cdots\!-\!b_{n},\quad  b_i\!\in\!\mathbb{R}_+,  \label{13a}   
\end{equation}  
with  the gcd of the indices $i$ of coefficients $b_i\!\neq\!0$ equal to 1, 
  are asymptotically simple \cite[Theorem 12.2]{Ostrowski}.  
Thus, for any given $p\!\in\!\mathbb{R}_+$ and $n$, the characteristic polynomial  
\begin{equation} 
  P^{(n)}_p(\lambda)\!\equiv\!\lambda^{n}\!-\!p\, (\lambda^{n-1}\!+\!\cdots\!+\! 1) \label{13}   
\end{equation}  
of the linear recurrence  
\begin{equation}  
  F^{(n)}_k(p)\!\equiv\!p\big(F^{(n)}_{k-1}(p)+\!\cdots\!+\! F^{(n)}_{k-n}(p)\big),\,\,  n\!\leq\!k,\quad \textrm{and} \quad F^{(n)}_{k\!}(p)\!=\!a_k\!\in\! \mathbb{R},\,\, 0\!\le\! k \!&lt;\! n, \label{11a} 
\end{equation} 
 has the unique simple positive dominant zero $\lambda^{(n)}(p)$, i.e., all other zeros of polynomial \eqref{13} have moduli  strictly smaller than $|\lambda^{(n)}(p)|$.  
 
In 1997, Dubeau et al.~\cite {Dubeau2} proved that, if the characteristic polynomial of an arbitrary linear recurrence 
is asymptotically simple, the limit of the ratios of the successive terms of the (not necessarily integer) sequence generated by the recurrence exists for at least one set of the initial conditions $\big\{a_{n-1}\!=\!1, a_k\!=\!0 \,|\,0\!\le\!k\!&lt;\!n\!-\!1\big\}$. Moreover, they showed that, if this limit exists for a given  set of initial conditions, the limit  coincides with the dominant zero of the characteristic polynomial, i.e.,  for a given $p$, $n$, and  $\big\{a_k\!\in\!\mathbb R \,|\,0\!\le\!k\!&lt;\!n\big\}$,  
\begin{equation} 
  \Phi^{(n)}(p)\!\equiv\!\lim_{k\to\infty} F^{(n)}_{k+1}(p)/F^{(n)}_{k}(p)\!=\!\lambda^{(n)}(p),\quad k &gt; k_0.  \label{20} 
\end{equation} 
In particular, for a given $m$ and $n$, the  $(m,n)$-anacci constant\,  $\Phi^{(n)}(m)$ is  equal to the dominant zero $\lambda^{(n)}(m)$ of the polynomial\, $P^{(n)}_m(\lambda)\!\equiv\!\lambda^{n}\!-\!m(\lambda^{n-1}\!+\!\cdots\!+\!1)$.  
 
\smallskip 
We derive the analytic representation of the set $\big\{\Phi^{(n)}(m)\,|\, m,n\!\in\!\mathbb{N}\big\}$  of the $(m,n)$-anacci constants  by proving there exist a continuous function \,$\overline{\mathbb{R}}_+^2\!\ni\!(p,q)\!\to\!\lambda(p,q)\!\in\!\overline{\mathbb{R}}_+$\, such that:  
\begin{enumerate} 
\item[a.] for any $p$ and $n$, \,$\lambda(p,n)\!=\!\lambda^{(n)}(p)$, i.e., in particular, $\lambda(m,n)\!=\!\Phi^{(n)}(m)$;  
 \item[b.] for any $(p,q)\!\in\!\mathbb{R} _+^2$ such that  $p\cdot q\!\neq\!1$, $\lambda(p,q)$ is of class $C^{\infty}$; 
\item[c.] the restriction $\lambda(p,q)|_\ell$ of $\lambda(p,q)$ to any half-line $\ell\!\subset\!{\mathbb{R}}_+^2$ is  strictly increasing;  
\item[d.] for any $p\!&gt;\!0$ and $q\!\ge\!1$,\, $p\!\leq\!\lambda(p,q)\!&lt;\!p\!+\!1$,  in particular, $m\!\leq\!\Phi^{(n)}(m)\!&lt;\!m\!+\!1$;  
\item[e.] for any $p\!\in\!\mathbb{R}_+,$\, $\lim_{q\to\infty}\lambda(p,q)\!=\!p\!+\!1$,  in particular, $\lim_{n\to\infty}\Phi^{(n)}(m)\!=\!m\!+\!1$. 
\end{enumerate} 
 
\smallskip 
The latter  result generalizes  the well-known fact regarding the limit of the $n$-anacci constants sequence: $\lim_{n\to\infty}\Phi^{(n)}(1)\!=\!2$, cf.,~e.g., \cite{Dubeau1, Flores}.  
 
Moreover, the result stated  in  (d) implies  that the set of the $(m,n)$-anacci constants  is totally ordered  as follows: if $n_2\!&gt;\!n_1$, then \,$\Phi^{(n_2)}(m_2)\!&gt;\!\Phi^{(n_1)}(m_1)$\, for any $m_2$ and $m_1$, whereas $\Phi^{(n)}(m_2)\!&gt;\!\Phi^{(n)}(m_1)$\, if $m_2\!&gt;\!m_1$.  
 
We have shown  \cite{Szczyrba}  how to construct geometric representations of the $(m,n)$-anacci constants correlated with this order using  dilations of convex compact sets in $n$-dimensional Euclidean spaces. 

\section{Analytic representation of the  ({\it m,n})-anacci constants }
 
The limits $\Phi^{(n)}(p)$, $p\!\in\!\mathbb R_+$, are also zeros of the  polynomials   
\begin{equation} 
Q^{(n)}_{\,p}(\lambda)\!\equiv\!\lambda^{n+1}\!-\!(p\!+\!1)\lambda^n\!+\!p\!=\!(\lambda-1)P^{(n)}_p(\lambda). \label{21} 
\end{equation} 
 We derive the analytic representation of the set $\big\{\Phi^{(n)}(p)\,|\, p\!\in\!\mathbb{R}_+,\, n\!\in\!\mathbb{N}\big\}$ using  the  function  
\begin{equation} 
 Q(\lambda,p,q)\!\equiv\!\lambda^{q+1}\!-\!(p\!+\!1)\lambda^q\!+\!p,\quad  \lambda, p, q\!\in\!\mathbb{R}_+. \label{22}  
\end{equation}  
The function $Q(\lambda,p,q)$ equals 0 at the plane $\lambda\!=\!1$ and at the zeros $\lambda^{(n)}(p)$. In particular, the restriction $Q(\lambda,1,q)$ of \,$Q(\lambda,p,q)$ to  the plane $p\!=\!1$ includes the sequence of the $n$-anacci constants $\big(\Phi^{(n)}(1)\big)_{n=1}^\infty$. 
 
\smallskip 
Figure \ref{Fig.1} depicts the restriction  $Q(\lambda,1,q)$ and the zero function \,$O(\lambda,q)\!\equiv\!0$.  The functions   intersect along  the zero line $O(1,q)$ and the zero curve, say  $\lambda_1(q)\!=\!0$, that is defined implicitly by the equation $Q(\lambda,1,q)\!=\!0$ and that includes the sequence $\big(\Phi^{(n)}(1)\big)_{k=1}^\infty$. 
 
\begin{figure}[H] 
  \centering
  \includegraphics[width=5.67in,height=2.36in,keepaspectratio]{1} 
  \caption{The  restriction $Q(\lambda,1,q)$  of the function $Q(\lambda,p,q)$ and the function $O(\lambda,q)\!\equiv\!0$, $0\!&lt;\!\lambda\!\leq\!2$ and  $0\!&lt;q\!\leq\!4$,  intersecting along  the zero line $O(1,q)$  and  the zero curve  $\lambda_1(q)$.  The locations of  $n$-anacci constants  $\Phi^{(n)}(1)$ with $1\!\le\!n\!\le\!4$ are marked by white ovals.} \label {Fig.1}
\end{figure} 
 
The equation  $Q(\lambda,a,q)\!=\!0$, $a\!\in\!\mathbb R_+$, defines the zero curve $\lambda_a(q)$. If $a\!=\!m\!\in\!\mathbb N$, the zero curve $\lambda_m(q)$  contains the sequence of the $(m,n)$-anacci constants  $\big(\Phi^{(n)}(m)\big)_{n=1}^\infty$.     
  
\medskip 
The next two theorems establish the analytic representation of  the set of the $(m,n)$-anacci constants $\big\{\Phi^{(n)}(m)\,|\, m,n\!\in\!\mathbb{N}\big\}$  as well as of all roots of  the function $Q(\lambda,p,q)$.  
 
\begin{theorem}  
\label{Thm.1} 
For any given $p,q\!\in\!\mathbb{R}_+$, $p\cdot q\!\neq\! 1$, the function $Q(\lambda ,p,q)$ of one variable $\lambda\!\in\!\mathbb R_+$ has the unique root $\lambda(p,q)\!\neq\!1$, whereas if\, $p\cdot q\!=\! 1$, its unique root $\lambda(p,1/p)\!=\!1$. Moreover, 
\begin{equation} 
1\!&lt;\!(p+1)\,q/(q+1\!)&lt;\!\lambda(p,q)  \quad \text{iff}\quad  p\cdot q\!&gt;\!1,  \label{23} 
\end{equation} 
\begin{equation} 
0\!&lt;\!\lambda(p,q)\!&lt;\!(p+1)\,q/(q+1)\!&lt;\!1  \quad \text{iff}\quad  p\cdot q\!&lt;\!1, \label{24} 
\end{equation} 
\begin{equation} 
\text{and} \quad \lambda(p,q)\!&lt;\!p\!+\!1 \quad \text{for any} \quad q\!\in\!\mathbb R_+. \label{25} 
\end{equation} 
\end{theorem}  
 
\begin{proof}  
The partial derivative of function \eqref{22} with respect to $\lambda$ is given by 
\begin{equation} 
 \partial Q(\lambda,p,q)/\partial\lambda\!=\!\lambda^{q-1}\big(\lambda(q\!+\!1)\!-\!(p\!+\!1)q\big). \label{26} 
\end{equation} 
Thus, for any $p,q\!\in\!\mathbb{R}_+$, the function $Q(\lambda,p,q)$ of the variable $\lambda\!\in\!\mathbb R_+$  has one local minimum at the point 
\begin{equation} 
\lambda_{min}(p,q)\!=\!(p\!+\!1)q/(q\!+\!1). \label{27} 
\end{equation}   
Formula  \eqref{27} implies that the minimum  is assumed at $\lambda_{min}\!=\!1$\, iff\, $p\cdot q\!=\!1$. Because, function \eqref{22}  equals zero at $\lambda\!=\!1$, therefore, if $p\cdot q\!=\!1$, 1 is the only root of the function $Q(\lambda,p,q)$  of the variable $\lambda$,  cf.~the left most white  oval \,$\Phi^{(1)}(1)\!\!=\!1$ \,in Figure \ref{Fig.1}.  
 
If\, $p\cdot q\!\neq\!1$, there exists a second {\em positive\/} root\, $\lambda(p,q)$  of \,$Q(\lambda,p,q)$\, besides 1 (if $\lambda_{min}\!&lt;\!1$, the  existence of the positive root is implied by the fact that $Q(0,p,q)\!=p\!&gt;\!0$), cf.~Figure \ref{Fig.1}.  
 
Moreover,  for any $p$ and $q$, the following holds: 
\begin{equation} 
 1\!&lt;\!\lambda_{min}(p,q)\!&lt;\!\lambda(p,q)\,\,\,\text{iff}\,\,\, p\cdot q\!&gt;\!1,\,\label{28} 
\end{equation} 
\begin{equation} 
\text{and if}\,\,\, p\cdot q\!&gt;\!1, \,\, 
 Q(\lambda,p,q)\!&lt;\!0\,\,\, \text{iff}\,\,\, 1\!&lt;\!\lambda&lt;\!\lambda(p,q);\label{28a} 
\end{equation} 
\begin{equation} 
 0\!&lt;\!\lambda(p,q)\!&lt;\!\lambda_{min}(p,q)\!&lt; \!1\,\,\, \text{iff}\,\,\, p\cdot q\!&lt;\!1,\, \label{29}\ 
\end{equation} 
\begin{equation} 
\text{and if}\,\,\, p\cdot q\!&lt;\!1,\,\, Q(\lambda,p,q)\!&lt;\!0\,\,\,\text{iff}\,\,\, \lambda(p,q)\!&lt;\!\lambda\!&lt;\!1; \label{29a}\ 
\end{equation} 
\begin{equation} 
\lambda(p,q)\!=\!\lambda_{min}(p,q)\!=\!1\,\,\, \text{iff}\,\,\, p\cdot q\!=\!1, \,  \label{210} 
\end{equation} 
 \begin{equation} 
\text{and  if}\,\,\, p\cdot q\!=\!1, 
\,\, Q(\lambda,p,q)\!&gt;\!0\,\,\,\text{iff 
}\,\, \lambda\!\neq\! 1.\label{210a} 
\end{equation} 
 
\smallskip 
Since we have  $Q(p\!+\!1,p,q)\!=\!p\!&gt;\!0$, formulas \eqref{28}--\eqref{210a} imply  that $\lambda(p,q)\!&lt;\!p\!+\!1$ for any $q\!\in\! \mathbb R_+$.  
 
\end{proof} 
 
\begin{theorem} 
\label{Thm.2} 
\begin{enumerate} 
\item[(i)] The assignment $\mathbb{R}_+^2\!\ni\!(p,q)\to\lambda(p,q)\!\in\!\mathbb{R}_+$ defines a continuous function 
 such that, for any $(p,n)\!\in\!\mathbb{R}_+\!\times\!\mathbb{N}$,  $\lambda(p,n)\!=\!\lambda^{(n)}(p)$ holds, i.e.,  $\lambda(m,n)\!=\!\Phi^{(n)}(m)$; 
 
\item[(ii)] \,if $p\cdot q\!\neq\!1$, the function $\lambda(p,q)$ is of class $C^{\infty}$\!;  
 
\item[(iii)] the restriction $\lambda(p,q)|_\ell$ of $\lambda(p,q)$ to any half-line $\ell\!\subset\!{\mathbb{R}}_+^2$  is strictly increasing;  
 
\item[(iv)] for any $p\!&gt;\!0$ and $q\!\ge\!1$, $p\!\leq\!\lambda(p,q)$,   i.e., $m\!\leq\!\Phi^{(n)}(m)\!&lt;\!m\!+\!1$;   
 
\item[(v)] for any $p\!\in\!\mathbb{R}_+,$ $\lim_{q\to\infty}\lambda(p,q)=p+1$, i.e., $\lim_{n\to\infty}\Phi^{(n)}(m)\!=\!m\!+\!1$;  
 
\item[(vi)]  the  plane ${\mathbb{R}}_+^2\!\ni(p,q)\!\to\!\mathcal P(p,q)\!\equiv\!p\!+\!1$  majorizes the function $\lambda(p,q)$  from above and is the   asymptotic  plane for $\lambda(p,q)$;  
 
\item[(vii)] for any $p_0\!\in\!\mathbb{R}_+$,  $\lim_{(p,q)\to(p_0,0)}\!\lambda(p,q)\!=\!0$, and for any $ q_0\!\in\!\mathbb{R}_+$, $\lim_{(p,q)\to(0,q_0)}\!\lambda(p,q)\!=\!0$, 
 i.e., the open domain ${\mathbb{R}}_+^2$ of $\lambda(p,q)$ can be extended to the closed domain $\overline{\mathbb{R}}_+^2$. 
\end{enumerate} 
\end{theorem} 
 
\begin{proof}  
(i) It follows from Theorem \ref{Thm.1}  that the  assignment  defines a function.   If, for $(p_0,q_0)$ with $p_0\cdot q_0\!=\!1$, $\lim_{(p,q)\to(p_0,q_0)}\!\lambda(p,q)\!\neq\!1\!=\!\lambda(p_0,q_0)$, then we have a contradiction with  \eqref{210a} due to the continuity of the function $Q(\lambda,p,q)$ and the fact that  $Q\big(\lambda(p,q),p,q\big)\!=\!0$.  
 
Thus, $\lambda(p,q)$ is continuous at $(p_0,q_0)$ with $p_0\cdot q_0\!=\!1$. The continuity of $\lambda(p,q)$ at $(p_0,q_0)$ with $p_0\cdot q_0\!\neq\!1$ is implied by part (ii). The definition of  $\lambda(p,q)$  assures that  $\lambda(p,n)\!=\!\lambda^{(n)}(p)$. 
 
\medskip 
(ii) The equation $Q\big(\lambda(p,q),p,q\big)\!=\!0$ defines the function $\lambda(p,q)$ implicitly.  
It follows from  formulas \eqref{26} and \eqref{27} that the partial derivative $\partial Q(\lambda,p,q)/\partial\lambda$ is continuous and equals 0 iff\,  $\lambda(p,q)\!=\!\lambda_{min}(p,q)$, i.e.,  according to \eqref{210a} iff\, $p \cdot q\!=\!1$.  
 
Thus, the implicit function theorem implies that, if\, $p\cdot q\!\neq\!1$, the function  $\lambda(p,q)$ is continuously differentiable and the following holds:  
\begin{equation} 
\frac{\partial\lambda(p,q)}{\partial p}\!=\!\frac{-1}{\frac{\partial Q\big(\lambda(p,q),p,q\big)}{\partial\lambda}}\cdot \frac{\partial Q\big(\lambda(p,q),p,q\big)}{\partial p}\!=\!\frac{1-\big(\lambda(p,q)\big)^q}{\left[\lambda(p,q)(q\!+\!1)\!-\!(p\!+\!1)q\right]\big(\lambda(p,q)\big)^{q-1}}, \label{211} 
\end{equation}  
\begin{equation} 
\frac{\partial\lambda(p,q)}{\partial q}\!=\!\frac{-1}{\frac{\partial Q\big(\lambda(p,q),p,q\big)}{\partial\lambda}}\cdot \frac{\partial Q\big(\lambda(p,q),p,q\big)}{\partial q}\!=\!\frac{\left[p+1-\lambda(p,q)\right]\big(\lambda(p,q)\big)^{q}\ln q} {\left[\lambda(p,q)(q\!+\!1)\!-\!(p\!+\!1)q\right]\big(\lambda(p,q)\big)^{q-1}}. \label{212} 
\end{equation}  
Since the function $\lambda(p,q)$ is continuously differentiable if $p\,\cdot\, q\!\neq\!1$, it follows from formulas \eqref{211} and  \eqref{212} that all partial derivatives of $\lambda(p,q)$ of an arbitrary  order  
exist and are continuous. Consequently, the function   $\lambda(p,q)$ is of class $C^{\infty}$ if\, $p\cdot q\!\neq\!1$.    
 
\medskip (iii) 
If $p\cdot q\!&gt;\!1$ (respectively $p\cdot q\!&lt;\!1$), the denominator and, according to formulas \eqref{23}--\eqref{25}, both numerators in  \eqref{211} and \eqref{212} are positive (respectively negative).  
Thus, the directional derivative of  $\lambda(p,q)$ 
along $\ell$  is positive, i.e., $\lambda(p,q)|_\ell$ is strictly increasing if $p\cdot q\!\neq\!1$.   
It follows from Theorem \ref{Thm.1} that  $\lambda(p,q)|_\ell$ is also strictly increasing  at  $p\cdot q\!=\!1$.  
 
\medskip  
(iv) Formula \eqref{13} implies that  \,$\lambda(p,1)\!=\!p$, which is smaller than $\lambda(p,q)$, $q\!&gt;\!1$, since for a fixed  $p$,  $\lambda(p,q)$ is strictly increasing.   
 
\medskip 
(v) The convergence of $\lambda(p,q)$ to $p\!+\!1$ follows from   \eqref{23} and \eqref{25}. 
 
\medskip  
(vi) The assertion follows from part (v). 
 
\medskip 
(vii) The first limit equals 0 due to formulas  \eqref{27} and \eqref{29}. Definition \eqref{22} implies that the second limit is equal to either 0 or 1. The latter is impossible due to \eqref{27} and \eqref{29}.   
 
\end{proof} 
 
The lower bound $(p\!+\!1)\,q/(q\!+\!1)$ for $\lambda(p,q)$ in \eqref{23}  predicts, e.g., that the golden ratio  $\Phi\!\equiv\!\lambda(1,2)$ is just greater than 4/3. In the case where $p\!=\!1$ and $q\!=\!n\!\in\!\mathbb N$, Wofram \cite{Wolfram} showed that $2(1\!-\!1/2^n)\!&lt;\!\lambda(1,n)$,  which implies that the $n$-anacci constants  $\Phi^{(n)}(1)\!=\!\lambda(1,n)$ are close to the limit 2 already when  $n$ is small. 
 
The next lemma provides  a lower bound for $\lambda(p,q)$ that is independent  
of q,  which shows that, for progressively larger values of  $p$, the roots $\lambda(p,q)$  are  closer and closer to the limit $p\!+\!1$. Consequently, when the  weight $m$ increases,   the $(m,n)$-anacci constants $\Phi^{(n)}(m)$ become closer and  closer to the limit $m\!+\!1$. 
 
\begin{lemma} \label{Thm.3} 
For any $q\!\ge\!2$  and   $p\!&gt;\!1/\Phi$,  the following holds: 
\begin{equation} 
p+\!1\!-\!1/(p+\!1)\!&lt;\!\lambda(p,q). \label{213}   
\end{equation} 
 Moreover, the lower bounds \eqref{23} and \eqref{213} satisfy 
\begin{equation} 
 (p+\!1)\,q/(q+\!1)\!\leq\! p+\!1\!-\!1/(p+\!1) \quad\text{iff}\quad q\!\leq\!(p+1)^2\!-\!1. \label{214} 
\end{equation}  
\end{lemma} 
 
\begin{proof} 
 Since\, $Q^{(2)}_p\big(p\!+\!1\!-\!1/(p\!+\!1)\big)\!=\!-p\,(p^2\!+\!p\!-1)/(p\!+\!1)^2\!&lt;\!0$\,  if \,$p\!&gt;\!1/\Phi$, formula  \eqref{28} implies  that  $p\!+\!1\!-\!1/(p\!+\!1)\!&lt;\!\lambda(p,2)$. For $q\!&gt;\!2$, $\lambda(p,2)\!&lt;\!\lambda(p,q)$ since $\lambda(p,q)$ is strictly increasing for a fixed $p$, cf.~part (iii) of Theorem \ref{Thm.2}. Formula \eqref{214} follows from a simple calculation. 
 
\end{proof} 
 
Figure \ref{Fig.2} shows the restrictions  $\lambda(a,q)$ of $\lambda(p,q)$ to the lines $p\!=\!a$, which are the same as  the zero curves $\lambda_a(q)$ introduced  above.   
 Each restriction  starts at $\lambda(a,1)\!=\!a$, increases asymptotically to $a\!+\!1$, and is less than $1/(a\!+\!1)$ from $a\!+\!1$  in the region to the right of the  line  $q\!=\!2$. The restrictions $\lambda(m,q)$, $m\!\in\!\mathbb N$,  include the  $(m,n)$-anacci constants  $\Phi^{(n)}(m)$.  

\begin{figure}[H]
  \centering
  \includegraphics[width=5.67in,height=2.4in,keepaspectratio]{2}
  \caption{The restrictions $\lambda(a,q)$,   $a\!=\!\frac{2}{3},1,\dots,2\frac{2}{3}$, $1\!\leq\!q\!\leq\!4$, of the function  $\lambda(p,q)$. The locations of  the $(m,n)$-anacci constants  $\Phi^{(n)}(m)$, $m\!=1,2$, $1\!\le\!n\!\le\!4$, are marked by white ovals.   The lower bound \eqref{213} exceeds the lower bound  \eqref{23} in the region  above the white curve $q\!=\!(p\!+\!1)^2\!-\!1$ that  goes through the points  ($3,1)$ and  $(4,2/\Phi)$.}  \label{Fig.2}
\end{figure} 
 
The function $\lambda(p,q)$ is also defined implicitly by the continuous function  
\begin{equation} 
p(\lambda,q)\!\equiv\!\lambda^q(\lambda\!-\!1)/(\lambda^q\!-\!1) \quad \text{if}\quad   \lambda\!\ne\!1 \quad \text{and} \label{215} 
\end{equation} 
\vskip -.1in 
\begin{equation} 
  p(\lambda,q)\!\equiv\!1/q \quad\text{if}\quad  \lambda\!=\!1,\label{216}  
\end{equation} 
 which  is  of class \,$C^{\infty}$ if $\lambda\!\neq\!1$. 
For $q\!=\!n\!\in\!\mathbb N$, the function defined by  \eqref{215}--\eqref{216}  takes the form   
\begin{equation} 
p(\lambda,n)=\frac{\lambda^n}{\Sigma_{k=0}^{n-1}\,\lambda^k}.\label{217}   
\end{equation} 
 
\smallskip  
Figure \ref{Fig.3} depicts the asymptotic  plane $\mathcal P(p,q)\!\equiv\!p+\!1$ and the function $\lambda(p,q)$ generated by   formulas \eqref{215} and \eqref{216}. The thick curves going up the graph of the function $\lambda(p,q)$ are the restrictions $\lambda(a,q)$  with  $a\!=\!\frac{1}{3},\frac{2}{3}\dots,2\frac{2}{3},3$.  They include the $(m,n)$-anacci constants  $\Phi^{(n)}(m)$,  $1\!\le\!m\!\le\!3$, $1\!\le\!n\!\le\!4$.  The thin curves increasing from left to right are the restrictions  $\lambda(p,q)|_\ell$. The horizontal thin curves are the level curves  $\lambda(p,q)\!=\!c$ with $c\!=\!\frac{1}{2},1\dots,3\frac{1}{2}, 4$. 
 
\begin{figure}[h] 
  \centering
  \includegraphics[width=5.67in,height=1.99in,keepaspectratio]{3}
  \caption{ The function $\lambda(p,q)$   and  the asymptotic plane $\mathcal P(p,q)$, $0\!\leq\!p\!\leq\!3$,  $0\!\leq\!q\!\leq\!4.1$.  } \label{Fig.3} 
\end{figure} 
 
 
Theorems \ref{Thm.1} and \ref{Thm.2} together with    formulas \eqref{213} and  \eqref{217} imply the following properties of the $(m,n)$-anacci constants $\Phi^{(n)}(m)$.
 
\begin{theorem} \label{Thm.4} 
\begin{enumerate} 
\item[(i)] The  sequences \,$\big(\Phi^{(n)}(m)\big)_{n=1}^{\infty}$  with a fixed   $m\!\in\!\mathbb{N}$  and  $\big(\Phi^{(n)}(m)\big)_{m=1}^{\infty}$ with a fixed $n\!\in\!\mathbb{N}$,  as well as the sequences\, $\big(\Phi^{(n)}(kn)\big)_{n=1}^{\infty}$\, and \,$\big(\Phi^{(km)}(m)\big)_{m=1}^{\infty}$ \,with a fixed  
 $k\!\in\!\mathbb{N}$ are strictly  increasing.  
 
\item[(ii)]  For any $n\!\in\!\mathbb{N}$, the sequence $\big(\frac{m\!+\!1}{m}\,\Phi^{(n)}(m)\big)_{m=1}^{\infty}$  is strictly increasing. 
 
\item[(iii)] If $n\!&gt;\!1$, 
the sequence $\big(\frac{1}{m}\,\Phi^{(n)}(m)\big)_{m=1}^{\infty}$ is strictly decreasing to $1$. If $n\!=\!1$, $\frac{1}{m}\,\Phi^{(1)}(m)\!=\!1$ for any $m$.  
 
\item[(iv)] The triple $\big(\lambda(p,q),p,q\big)\!\in\!\mathbb{N}^3$ \,iff\, $\big(\lambda(p,q),p,q\big)\!=\!(m,m,\!1)$,  $m\!\in\!\mathbb{N}$, i.e.,  
the  $(m,n)$-anacci constants\, $\Phi^{(n)}(m)$ are integer iff \,$n\!=\!1$.  
 
 
\item[(v)]  If the function $\lambda(p,n)\!=\!m\!\in\!\mathbb{N}$\,  for some\, $1\!&lt;\!n\!\in\!\mathbb N$,  then  $p$ is rational and  
 $(m\!-\!1)\!&lt;\!p\!&lt;\!m$. 
 
\end{enumerate} 
\end{theorem} 
\begin{proof} (i) The assertions follow from part (iii) of Theorem \ref{Thm.2}. 
 
\medskip 
(ii) If $n\!=\!1$,  the sequence $\big(\frac{m+1}{m}\,\Phi^{(1)}(m)\big)_{\!m=1}^{\!\infty}\!=\!m\!+\!1$   since\, $\Phi^{(1)}(m)\!=\!m$ according to definitions \eqref{11} and \eqref{12}.  
 If $n\!&gt;\!1$,   formulas  \eqref{25} and \eqref{213} imply that   
\begin{equation} 
 m\!+\!1\!-\!1/(m\!+\!1)\!&lt;\!\Phi^{(n)}(m)\!&lt;\!m\!+\!1. \label{218}  
\end{equation}  
 Thus, if $n\!&gt;\!1$, 
  \begin{equation} 
\frac{m\!+\!1}{m}\Phi^{(n)}(m)\!&lt;\!\frac{(m\!+\!1)^2}{m}\!\leq\!\frac{\!m+\!2}{m\!+\!1}\big(m\!+\!2\!-\!\frac{1}{(m\!+\!2)}\big)\!&lt;\!\frac{m\!+\!2}{m\!+\!1}\,\Phi^{(n)}(m\!+\!1) \label{219}  
\end{equation} 
 is true for any $m$, due to formula  \eqref{218} and the fact that the middle inequality in formula \eqref{219} reduces to $m\!\ge\!1$. 
 
\medskip 
(iii) If $n\!&gt;\!1$, the following inequalities hold due to formula \eqref{218}: 
\begin{equation} 
\frac{m\!+\!2}{m\!+\!1}\!-\!\frac{1}{(m\!+\!2)(m\!+\!1)}\!&lt;\!\frac{\Phi^{(n)}(m\!+\!1)}{m\!+\!1}\!&lt;\!1\!+\!\frac{1}{m\!+\!1}\!&lt;\!\frac{\Phi^{(n)}(m)}{m}\!&lt;\!1\!+\!\frac{1}{m}. \label{220} 
\end{equation} 
 
\medskip 
(iv) The \emph{only if} part of the assertion  follows from formula \eqref{218}. 
 
\medskip 
(v) The assertion is implied by formula \eqref{217} and the fact that $p\!&lt;\!\lambda(p,n)\!=\!m\!&lt;\!p+\!1$ for $n\!&gt;\!1$,  due to formulas \eqref{25} and \eqref{213},  cf.~also Figure \ref{Fig.2}. 
 
\end{proof}  

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\bibitem{Dubeau2} F.~Dubeau et al., {On weighted $r$-generalized Fibonacci sequences}, \emph{Fibonacci Quart.}~\textbf{35} (1997), 102--110. 
 
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\bibitem{Horadam1}A.~F.~Horadam, {A generalized Fibonacci sequence}, \emph{Amer.~Math.~Monthly} \textbf{68} (1961), 455--459. 
 
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\end{thebibliography}

\bigskip \hrule \bigskip

\noindent 2010 {\it Mathematics Subject Classification}:  Primary
11B37; Secondary 11B39.

\noindent \emph{Keywords:} linear recurrence, $n$-step Fibonacci
number, weighted $n$-generalized Fibonacci sequence,  generalized
$n$-anacci constant.

\bigskip \hrule \bigskip

\noindent (Concerned with sequences 
\seqnum{A000012},
\seqnum{A000032},
\seqnum{A000045},
\seqnum{A000073},
\seqnum{A000078},
\seqnum{A000213},
\seqnum{A000288},
\seqnum{A000322},
\seqnum{A001590},
\seqnum{A001591},
\seqnum{A001592},
\seqnum{A001630},
\seqnum{A001644},
\seqnum{A002605},
\seqnum{A007486},
\seqnum{A010924},
\seqnum{A015577},
\seqnum{A020992},
\seqnum{A021006},
\seqnum{A023424},
\seqnum{A026150},
\seqnum{A028859},
\seqnum{A028860},
\seqnum{A030195},
\seqnum{A057087},
\seqnum{A057088},
\seqnum{A057089},
\seqnum{A057090},
\seqnum{A057092},
\seqnum{A057093},
\seqnum{A066178},
\seqnum{A073817},
\seqnum{A074048},
\seqnum{A074584},
\seqnum{A077835},
\seqnum{A079262},
\seqnum{A080040},
\seqnum{A081172},
\seqnum{A083337},
\seqnum{A084128},
\seqnum{A085480},
\seqnum{A086192},
\seqnum{A086213},
\seqnum{A086347},
\seqnum{A094013},
\seqnum{A100532},
\seqnum{A100683},
\seqnum{A104144},
\seqnum{A104621},
\seqnum{A105565},
\seqnum{A105754},
\seqnum{A105755},
\seqnum{A106435},
\seqnum{A106568},
\seqnum{A108051},
\seqnum{A108306},
\seqnum{A116556},
\seqnum{A121907},
\seqnum{A122189},
\seqnum{A122265},
\seqnum{A123620},
\seqnum{A123871},
\seqnum{A123887},
\seqnum{A124312},
\seqnum{A125145},
\seqnum{A134924},
\seqnum{A141036},
\seqnum{A141523},
\seqnum{A145027},
\seqnum{A164540},
\seqnum{A164545},
\seqnum{A164593},
\seqnum{A168082},
\seqnum{A168083},
\seqnum{A168084},
\seqnum{A170931},
\seqnum{A181140},
\seqnum{A214727},
\seqnum{A214825},
\seqnum{A214826},
\seqnum{A214827},
\seqnum{A214828}, and
\seqnum{A214899}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received October 10 2014;
revised version received  February 23 2015.
Published in {\it Journal of Integer Sequences},
May 13 2015.

\bigskip
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\noindent
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