Abstract
In this work, we define a quadra Fibona-Pell integer sequence \(W_{n}=3W_{n-1}-3W_{n-3}-W_{n-4}\) for \(n\geq4\) with initial values \(W_{0}=W_{1}=0\), \(W_{2}=1\), \(W_{3}=3\), and we derive some algebraic identities on it including its relationship with Fibonacci and Pell numbers.
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1 Preliminaries
Let p and q be non-zero integers such that \(D=p^{2}-4q\neq0\) (to exclude a degenerate case). We set the sequences \(U_{n}\) and \(V_{n}\) to be
for \(n\geq2\) with initial values \(U_{0}=0\), \(U_{1}=1\), \(V_{0}=2\), and \(V_{1}=p\). The sequences \(U_{n}\) and \(V_{n}\) are called the (first and second) Lucas sequences with parameters p and q. \(V_{n}\) is also called the companion Lucas sequence with parameters p and q.
The characteristic equation of \(U_{n}\) and \(V_{n}\) is \(x^{2}-px+q=0\) and hence the roots of it are \(x_{1}=\frac{p+\sqrt{D}}{2}\) and \(x_{2}=\frac{p-\sqrt{D}}{2}\). So their Binet formulas are
for \(n\geq0\). For the companion matrix \(M=\bigl[ {\scriptsize\begin{matrix} p & -q \cr 1 & 0\end{matrix}} \bigr] \), one has
for \(n\geq1\). The generating functions of \(U_{n}\) and \(V_{n}\) are
Fibonacci, Lucas, Pell, and Pell-Lucas numbers can be derived from (1). Indeed for \(p=1\) and \(q=-1\), the numbers \(U_{n}=U_{n}(1,-1)\) are called the Fibonacci numbers (A000045 in OEIS), while the numbers \(V_{n}=V_{n}(1,-1)\) are called the Lucas numbers (A000032 in OEIS). Similarly, for \(p=2\) and \(q=-1\), the numbers \(U_{n}=U_{n}(2,-1)\) are called the Pell numbers (A000129 in OEIS), while the numbers \(V_{n}=V_{n}(2,-1)\) are called the Pell-Lucas (A002203 in OEIS) (companion Pell) numbers (for further details see [1–6]).
2 Quadra Fibona-Pell sequence
In [7], the author considered the quadra Pell numbers \(D(n)\), which are the numbers of the form \(D(n)=D(n-2)+2D(n-3)+D(n-4)\) for \(n\geq4\) with initial values \(D(0)=D(1)=D(2)=1\), \(D(3)=2\), and the author derived some algebraic relations on it.
In [8], the authors considered the integer sequence (with four parameters) \(T_{n}=-5T_{n-1}-5T_{n-2}+2T_{n-3}+2T_{n-4}\) with initial values \(T_{0}=0\), \(T_{1}=0\), \(T_{2}=-3\), \(T_{3}=12\), and they derived some algebraic relations on it.
In the present paper, we want to define a similar sequence related to Fibonacci and Pell numbers and derive some algebraic relations on it. For this reason, we set the integer sequence \(W_{n}\) to be
for \(n\geq4\) with initial values \(W_{0}=W_{1}=0\), \(W_{2}=1\), \(W_{3}=3\) and call it a quadra Fibona-Pell sequence. Here one may wonder why we choose this equation and call it a quadra Fibona-Pell sequence. Let us explain: We will see below that the roots of the characteristic equation of \(W_{n}\) are the roots of the characteristic equations of both Fibonacci and Pell sequences. Indeed, the characteristic equation of (3) is \(x^{4}-3x^{3}+3x+1=0\) and hence the roots of it are
(Here α, β are the roots of the characteristic equation of Fibonacci numbers and γ, δ are the roots of the characteristic equation of Pell numbers.) Then we can give the following results for \(W_{n}\).
Theorem 1
The generating function for \(W_{n}\) is
Proof
The generating function \(W(x)\) is a function whose formal power series expansion at \(x=0\) has the form
Since the characteristic equation of (3) is \(x^{4}-3x^{3}+3x+1=0\), we get
Notice that \(W_{0}=W_{1}=0\), \(W_{2}=1\), \(W_{3}=3\), and \(W_{n}=3W_{n-1}-3W_{n-3}-W_{n-4}\). So \((1-3x+3x^{3}+x^{4})W(x)=x^{2}\) and hence the result is obvious. □
Theorem 2
The Binet formula for \(W_{n}\) is
for \(n\geq0\).
Proof
Note that the generating function is \(W(x)=\frac{x^{2}}{x^{4}+3x^{3}-3x+1}\). It is easily seen that \(x^{4}+3x^{3}-3x+1=(1-x-x^{2})(1-2x-x^{2})\). So we can rewrite \(W(x)\) as
From (2), we see that the generating function for Pell numbers is
and the generating function for the Fibonacci numbers is
From (5), (6), (7), we get \(W(x)=P(x)-F(x)\). So \(W_{n}= ( \frac{\gamma^{n}-\delta^{n}}{\gamma-\delta } ) - ( \frac{\alpha^{n}-\beta^{n}}{\alpha-\beta } )\) as we wanted. □
The relationship with Fibonacci, Lucas, and Pell numbers is given below.
Theorem 3
For the sequences \(W_{n}\), \(F_{n}\), \(L_{n}\), and \(P_{n}\), we have:
-
(1)
\(W_{n}=P_{n}-F_{n}\) for \(n\geq0\).
-
(2)
\(W_{n+1}+W_{n-1}=(\gamma^{n}+\delta^{n})-(\alpha^{n}+\beta^{n})\) for \(n\geq1\).
-
(3)
\(\sqrt{5}F_{n}+2\sqrt{2}P_{n}=(\gamma^{n}-\delta^{n})+(\alpha ^{n}-\beta^{n})\) for \(n\geq1\).
-
(4)
\(L_{n}+P_{n+1}+P_{n-1}=\alpha^{n}+\beta^{n}+\gamma^{n}+\delta^{n}\) for \(n\geq1\).
-
(5)
\(2(W_{n+1}-W_{n}+F_{n-1})=\gamma^{n}+\delta^{n}\) for \(n\geq1\).
-
(6)
\(\lim_{n\rightarrow\infty}\frac{W_{n}}{W_{n-1}}=\gamma\).
Proof
(1) It is clear from the above theorem, since \(W(x)=P(x)-F(x)\).
(2) Since \(6W_{n-1}+W_{n+2}=3W_{n+1}-3W_{n-1}-W_{n-2}+6W_{n-1}\), we get
since \(\frac{6}{\gamma}+\frac{1}{\gamma^{2}}+\gamma ^{2}=\frac{-6}{\delta}-\frac{1}{\delta^{2}}-\delta ^{2}=6\sqrt{2}\) and \(\frac{-6}{\alpha}-\frac{1}{\alpha ^{2}}-\alpha^{2}=\frac{6}{\beta}+ \frac{1}{\beta^{2}}+\beta ^{2}=-3\sqrt{5}\).
(3) Notice that \(F_{n}=\frac{\alpha^{n}-\beta^{n}}{\alpha-\beta}\) and \(P_{n}=\frac{\gamma^{n}-\delta^{n}}{\gamma-\delta}\). So we get \(\sqrt{5}F_{n}=\alpha^{n}-\beta^{n}\) and \(2\sqrt{2} P_{n}=\gamma ^{n}-\delta^{n}\). Thus clearly, \(\sqrt{5}F_{n}+2\sqrt{2}P_{n}=(\gamma^{n}-\delta^{n})+(\alpha ^{n}-\beta^{n})\).
(4) It is easily seen that \(P_{n+1}+P_{n-1}=\gamma^{n}+\delta^{n}\). Also \(L_{n}=\alpha^{n}+\beta^{n}\). So \(L_{n}+P_{n+1}+P_{n-1}=\alpha ^{n}+\beta^{n}+\gamma^{n}+\delta^{n}\).
(5) Since \(W_{n+1}=3W_{n}-3W_{n-2}-W_{n-3}\), we easily get
and hence
since \(\frac{2\gamma^{3}-3\gamma-1}{\gamma^{3}}=\frac{-2\delta ^{3}+3\delta+1}{\delta^{3}}=\sqrt{2}\) and \(\frac{2\alpha ^{3}-3\alpha-1}{\alpha^{2}}=\frac{2\beta^{3}-3\beta-1}{\beta ^{2}}=1\).
(6) It is just an algebraic computation, since \(W_{n}= ( \frac{\gamma^{n}-\delta^{n}}{\gamma-\delta} ) - ( \frac{\alpha^{n}-\beta^{n}}{\alpha-\beta} ) \). □
Theorem 4
The sum of the first n terms of \(W_{n}\) is
for \(n\geq3\).
Proof
Recall that \(W_{n}=3W_{n-1}-3W_{n-3}-W_{n-4}\). So
Applying (9), we deduce that
If we sum of both sides of (10), then we obtain \(W_{n-3}+W_{0}+2(W_{1}+\cdots+W_{n-4})=3(W_{3}+W_{4}+\cdots +W_{n-1})-2(W_{1}+W_{2}+\cdots+W_{n-3})-(W_{4}+W_{5}+\cdots +W_{n})\). So we get \(W_{n-3}+2(W_{1}+W_{2}+\cdots +W_{n-4})=1-W_{n-2}-W_{n-1}-W_{n}+3W_{n-2}+3W_{n-1}\) and hence we get the desired result. □
Theorem 5
The recurrence relations are
for \(n\geq4\).
Proof
Recall that \(W_{n}=3W_{n-1}-3W_{n-3}-W_{n-4}\). So \(W_{2n}=3W_{2n-1}-3W_{2n-3}-W_{2n-4}\) and hence
The other assertion can be proved similarly. □
The rank of an integer N is defined to be
Thus we can give the following theorem.
Theorem 6
The rank of \(W_{n}\) is
for an integer \(k\geq0\).
Proof
Let \(n=5+6k\). We prove it by induction on k. Let \(k=0\). Then we get \(W_{5}=24=2^{3}\cdot3\). So \(\rho(W_{5})=2\). Let us assume that the rank of \(W_{n}\) is 2 for \(n=k-1\), that is, \(\rho(W_{6k-1})=2\), so \(W_{5+6(k-1)}=W_{6k-1}=2^{a}\cdot B\) for some integers \(a\geq1\) and \(B>0\). For \(n=k\), we get
Therefore \(\rho(W_{5+6k})=2\). Similarly it can be shown that \(\rho(W_{6+6k})=\rho(W_{7+6k})=2\).
Now let \(n=8+12k\). For \(k=0\), we get \(W_{8}=387=3^{2}\cdot43\). So \(\rho(W_{8})=3\). Let us assume that for \(n=k-1\) the rank of \(W_{n}\) is 3, that is, \(\rho(W_{8+12(k-1)})=\rho (W_{12k-4})=3^{b}\cdot H\) for some integers \(b\geq1\) and \(H>0\) which is not even integer. For \(n=k\), we get
So \(\rho(W_{12k+8})=3\). The others can be proved similarly. □
Remark 1
Apart from the above theorem, we see that \(\rho(W_{22})=\rho (W_{26})=\infty\), while \(\rho(W_{70})=\rho(W_{98})=13\) and \(\rho (W_{10})=\rho(W_{34})=\rho(W_{50})=23\). But there is no general formula.
The companion matrix for \(W_{n}\) is
Set
and
Then we can give the following theorem, which can be proved by induction on n.
Theorem 7
For the sequence \(W_{n}\), we have:
(1) \(RM^{n}N=W_{n+3}+P_{n}+2(W_{n+1}-F_{n})\) for \(n\geq1\).
(2) \(R(M^{T})^{n-3}N=W_{n}\) for \(n\geq3\).
(3) If \(n\geq7\) is odd, then
where
and if \(n\geq8\) is even, then
where
A circulant matrix is a matrix \(A=[a_{ij}]_{n\times n}\) defined to be
where \(a_{i}\) are constants. The eigenvalues of A are
where \(w=e^{\frac{2\pi i}{n}}\), \(i=\sqrt{-1}\), and \(j=0,1,\ldots ,n-1\). The spectral norm for a matrix \(B=[b_{ij}]_{n\times m}\) is defined to be \(\|B\|_{\mathrm{spec}}=\max\{\sqrt{\lambda_{i}}\}\), where \(\lambda_{i}\) are the eigenvalues of \(B^{H}B\) for \(0\leq j\leq n-1\) and \(B^{H}\) denotes the conjugate transpose of B.
For the circulant matrix
for \(W_{n}\), we can give the following theorem.
Theorem 8
The eigenvalues of W are
for \(j=0,1,2,\ldots,n-1\).
Proof
Applying (11) we easily get
since \(\alpha\beta=-1\), \(\gamma\delta=-1\), \(\alpha+\beta=1\), \(\alpha-\beta=\sqrt{5}\), \(\gamma+\delta=2\), and \(\gamma-\delta =2\sqrt{2}\). □
After all, we consider the spectral norm of W. Let \(n=2\). Then \(W_{2}=[0]_{2\times2}\). So \(\|W_{2}\|_{\mathrm{spec}}=0\). Similarly for \(n=3\), we get
and hence \(W_{3}^{H}W_{3}=I_{3}\). So \(\|W_{3}\|_{\mathrm{spec}}=1\). For \(n\geq4\), the spectral norm of \(W_{n}\) is given by the following theorem, which can be proved by induction on n.
Theorem 9
The spectral norm of \(W_{n}\) is
for \(n\geq4\).
For example, let \(n=6\). Then the eigenvalues of \(W_{6}^{H}W_{6}\) are
So the spectral norm is \(\|W_{6}\|_{\mathrm{spec}}=\sqrt{\lambda_{0}}=37\). Also \(\frac{W_{5}+4W_{4}+4W_{3}+W_{2}+1}{2}=37\). Consequently,
as we claimed.
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Acknowledgements
The author wishes to thank Professor Ahmet Tekcan of Uludag University for constructive suggestions.
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Özkoç, A. Some algebraic identities on quadra Fibona-Pell integer sequence. Adv Differ Equ 2015, 148 (2015). https://doi.org/10.1186/s13662-015-0486-7
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DOI: https://doi.org/10.1186/s13662-015-0486-7