Complete Sequences of Weighted r-Generalized Fibonacci Powers

Passawan Noppakaew and Thanakorn Prinyasart^{1} Department of Mathematics Faculty of Science Silpakorn University Nakhon Pathom 73000 Thailand

               noppakaew p@silpakorn.edu
 prinyasart t@silpakorn.edu, thanakorn dpst@hotmail.com

Abstract

In this paper, we study the least number L ∈ ℕ that makes the L-fold of a weighted r-generalized Fibonacci power complete. We can establish an upper bound and a lower bound for L, depending on the first r terms and the limit of the consecutive ratios. We also give the explicit value of L in some special cases. In the end, we study a particular minimal complete sequence called a generalized distinguished sequence and show that there exists a generalized distinguished sequence of a weighted r-generalized Fibonacci mth power for all large m ∈ ℕ.

1 Introduction

In 1960, Hoggatt and King [5] defined a complete sequence (a_{n}) as a sequence of positive integers such that for each m ∈ ℕ there exist finite distinct natural numbers n_{1}, n_{2}, …, n_{k} such that m = ∑_{i=1}^{k} a_{n_{i}}. They also showed that the Fibonacci sequence (F_{n}), defined by the recurrence F_{0} = 0, F_{1} = 1, and F_{n} = F_{n−1} + F_{n−2} for n ≥ 2, is complete. Moreover, they proved that any deletion of an element from (F_{n}) results in a non-complete sequence.

1Thanakorn Prinyasart is the corresponding author.

In this case, we say that (F_{n}) is a minimal complete sequence. A necessary and sufficient condition for a sequence to be complete was given by Brown [1]. A nontrivial example of a non-complete sequence is the sequence of Fibonacci squares, which is 1, 1, 4, 9, 25, 64, . . .. However, O’Connell [7] showed that the 2-fold of the sequence of Fibonacci squares, which is 1, 1, 1, 1, 4, 4, 9, 9, 25, 25, 64, 64, …, is complete. Hunsucker and Wardlaw [6] considerably generalized the result of O’Connell by proving that, for each m ∈ ℕ, the 2^{m−1}-fold, but not (2^{m−1} − 1)-fold, of (F_{n}^{m}) is complete. In their work, they also provided an example of an increasing complete sequence, which is minimal, and mapped it onto {F_{n}^{m} : n ∈ ℕ}.

Our aim in this paper is to extend the results of Hunsucker and Wardlaw to the weighted rgeneralized Fibonacci sequences. As far as we know, these sequences were firstly described by Dubeau et al. [2]. Let c_{1}, c_{2}, …, c_{r}, where r ≥ 2, be positive integers and let A_{1}, A_{2}, …, A_{r} be nonnegative integers with A_{r} > 0. A weighted r-generalized Fibonacci sequence (with positive integral initial conditions and nonnegative coefficients) (a_{n}) is defined by

a_{n} := \begin{cases}c_{n}, & if n = 1, 2, …, r; \\ ∑_{k=1}^{r} A_{k}a_{n−k}, & if n > r. \end{cases}  (1)

We call c_{1}, c_{2}, …, c_{r} initial values and A_{1}, A_{2}, …, A_{r} coefficients of (a_{n}). Notice that when r = 2, and c_{1} = c_{2} = A_{1} = A_{2} = 1, this sequence is the usual well-known Fibonacci sequence. We are concerned with not only the completeness of the L-fold of a weighted r-generalized Fibonacci sequence but also a particular minimal sequence, called a distinguished sequence, of a weighted r-generalized Fibonacci power as introduced by Hunsucker and Wardlaw [6].

2 Weighted r-generalized Fibonacci sequences

According to Dubeau et al. [2], a weighted r generalized Fibonacci sequence (a_{n}) is defined as the sequence generated by the recurrence (1) with c_{k}, A_{k} ∈ ℂ for all k ∈ {1, 2, …, r} and A_{r} ≠ 0. The characteristic polynomial of (a_{n}) is p(x) := x^{r} − ∑_{k=1}^{r} A_{k}x^{r−k}. It is well-known that (a_{n}) can be written in an exponential polynomial form, a_{n} = ∑_{k=1}^{l} C_{k}(n)λ_{k}^{n}, where λ_{1}, λ_{2}, …, λ_{l} ∈ ℂ are all distinct roots of p(x) and C_{k}(x) ∈ ℂ[x] with deg(C_{k}(x)) + 1 ≤ multiplicity of λ_{k} for all k ∈ {1, 2, …, l}. This exponential polynomial form is convenient to use when we want to determine the limit of the consecutive terms of the sequence, which is one of the crucial ingredients we need to obtain most of our results. Because of the appearance of the roots of the characteristic polynomials in the exponential polynomial forms, it is useful to gather the results about these roots from the work of Dubeau et al. [2] and Ostrowski [8] in this section.

The first upcoming theorem provides a useful result on the largest modulus roots of p(x), when A_{k} ≥ 0 for all k ∈ {1, 2, …, r − 1} and A_{r} > 0, which is the case we are interested in.

Theorem 1. [8, Theorem 12.2] Let p(x) := x^{r} − ∑_{k=1}^{r} A_{k}x^{r−k} ∈ ℝ[x] for some integer r ≥ 2, and nonnegative real numbers A_{1}, A_{2}, …, A_{r} with A_{r} > 0. If gcd{k : A_{k} > 0} = 1, then p(x) has a unique largest modulus root in ℂ. Moreover, this root is a positive real root with multiplicity 1.

According to Dubeau et al. [2], a polynomial p(x) is called asymptotically simple if there exists a unique root λ among its roots of maximum modulus that has maximal multiplicity. The root λ is then called the dominant root and its multiplicity is called the dominant multiplicity. By Theorem 1, if (a_{n}) is a weighted r-generalized Fibonacci sequence generated by (1) with A_{k} ≥ 0 for all k ∈ {1, 2, …, r − 1}, A_{r} > 0, and gcd{k : A_{k} > 0} = 1, then its characteristic polynomial is asymptotically simple with a simple positive dominant root λ [2, Theorem 9]. Even though λ is a simple positive dominant root, it does not imply the existence of lim_{n→∞} \frac{a_{n+1}}{a_{n}} in general [4, Example 4.6]. However, if we assume further that the initial values c_{1}, c_{2}, …, c_{r} are nonnegative real numbers, not all zero, then we can show that lim_{n→∞} \frac{a_{n+1}}{a_{n}} = λ with the aid of the following theorem.

Theorem 2. [2, Theorem 5] Let A_{1}, A_{2}, …, A_{r} ∈ ℂ with A_{r} ≠ 0, and λ be an arbitrary nonzero complex number. We denote the sequence (a_{n}) generated by the recurrence relation (1) with initial values c := (c_{1}, c_{2}, …, c_{r}) ∈ ℂ^{r} by (a_{n}(c)). Fix c_{0} := (0, 0, …, 0, 1) ∈ ℂ^{r}. Then lim_{n→∞} \frac{a_{n}(c)}{λ^{n}} exists for all c ∈ ℂ^{r} if and only if lim_{n→∞} \frac{a_{n}(c_{0})}{λ^{n}} exists. Moreover, when all the limits exist,

lim_{n→∞} \frac{a_{n}(c)}{λ^{n}} = (c_{r} + ∑_{k=1}^{r−1} c_{r−k}λ^{k} ∑_{l=k}^{r−1} \frac{A_{1+l}}{λ^{1+l}} ) · lim_{n→∞} \frac{a_{n}(c_{0})}{λ^{n}}  (2)

for all c := (c_{1}, c_{2}, …, c_{r}) ∈ ℂ^{r}.

By the above theorem, we have the following result.

Proposition 3. Let (a_{n}) be a weighted r-generalized Fibonacci sequence defined by (1), with c_{k}, A_{k} ≥ 0 for all k ∈ {1, 2, …, r}, A_{r} > 0, and c_{k′} > 0 for some k′ ∈ {1, 2, …, r}. Assume that gcd{k : A_{k} > 0} = 1. If λ is the dominant root, which is a simple positive root of its characteristic polynomial, then there exists C > 0 such that

a_{n} = Cλ^{n} + o(λ^{n})

as n → ∞. Consequently, lim_{n→∞} \frac{a_{n+1}}{a_{n}} = λ.

Proof. Notice that, if c′ := (λ, λ^{2}, …, λ^{r}) ∈ ℝ^{r}, then a_{n}(c′) = λ^{n} for all n ∈ ℕ, and so lim_{n→∞} \frac{a_{n}(c′)}{λ^{n}} = lim_{n→∞} \frac{λ^{n}}{λ^{n}} = 1. Since λ is the dominant root, lim_{n→∞} \frac{a_{n}(c_{0})}{λ^{n}} exists, where c_{0} := (0, 0, …, 0, 1) ∈ ℝ^{r}. Moreover, by (2), since c′ ≠ 0 and A_{r} > 0, we have that lim_{n→∞} \frac{a_{n}(c_{0})}{λ^{n}} > 0. Again, by (2), since c := (c_{1}, c_{2}, …, c_{r}) ≠ 0 and A_{r} > 0, we obtain that lim_{n→∞} \frac{a_{n}}{λ^{n}} = lim_{n→∞} \frac{a_{n}(c)}{λ^{n}} > 0. Hence, there exists C > 0 such that a_{n} = Cλ^{n} + o(λ^{n}) as n → ∞.

3 Notation

In this section, we provide some notation that will be used in the following sections. From now on, the weighted r-generalized Fibonacci sequences that we are interested in have positive integral initial values and nonnegative coefficients. Since the completeness of a sequence is invariant under rearrangements, it is sometimes easier to consider its nondecreasing rearrangement to justify the completeness of the sequence. So, we define the following equivalence relation for later use.

Definition 4. Let (a_{n}) and (b_{n}) be sequences of real numbers. We say that (a_{n}) and (b_{n}) are equivalent, denoted by (a_{n}) ∼ (b_{n}), if there exists a bijection ρ : ℕ → ℕ such that a_{n} = b_{ρ(n)} for all n ∈ ℕ. We denote the equivalence class containing (a_{n}) by [a_{n}].

From Proposition 3, we see that if gcd{k : A_{k} > 0} = 1, then the weighted r-generalized Fibonacci sequence has a nice exponential polynomial form and the limit of consecutive terms exists. This leads us to give a name to the recurrence relation of this form.

Definition 5. Let (a_{n}) be defined by (1). We say that (a_{n}) is in a primitive form if gcd {k : A_{k} > 0} = 1.

The next notation will be handy in the last section when we study a sequence that may not be in a primitive form. Moreover, we can also use this notation to define an L-fold of any sequence.

Definition 6. For sequences (a_{n}) and (b_{n}) of real numbers, we define [a_{n}] ⊕ [b_{n}] to be the equivalence class containing the sequence (c_{n}) satisfying c_{2k−1} = a_{k} and c_{2k} = b_{k} for all k ∈ ℕ.

Definition 7. Let (a_{n}) be a sequence of real numbers. For L ∈ ℕ, we call each sequence in the equivalence class \underbrace{[a_{n}] ⊕ [a_{n}] ⊕ ⋯ ⊕ [a_{n}]}_{L times} an L-fold of (a_{n}).

If a weighted r-generalized Fibonacci sequence does not have a nondecreasing rearrangement, then this sequence is periodic and easy to understand as shown in the following theorem.

Theorem 8. Let (a_{n}) be a weighted r-generalized Fibonacci sequence. Then (a_{n}) does not have a nondecreasing rearrangement if and only if there exist s ≥ 2 distinct constant sequences (b_{n}^{(1)}), (b_{n}^{(2)}), …, (b_{n}^{(s)}) such that

[a_{n}] = [b_{n}^{(1)}] ⊕ [b_{n}^{(2)}] ⊕ ⋯ ⊕ [b_{n}^{(s)}].

Moreover, when this happens, a_{n} = a_{n−r} for all n > r and, for m ∈ ℕ, we have that (a_{n}^{m}) is complete if and only if a_{k} = 1 for some k ≤ r.

Proof. The converse of the theorem is obvious. Now, assume that (a_{n}) does not have a nondecreasing rearrangement. Since c_{i} ≥ 1 for all i ≤ r, if A_{k} > 0 for some k < r or A_{r} > 1, then lim_{n→∞} a_{n} = ∞. So (a_{n}) has a nondecreasing rearrangement, which is a contradiction. This implies that A_{k} = 0 for all k < r and A_{r} = 1. Thus, a_{n} = a_{n−r} for all n > r, and we are done.

Hence, from now on, we will consider only weighted r-generalized Fibonacci sequences that have a nondecreasing rearrangement.

Definition 9. A sequence of real numbers (a_{n}) is increasable if it has a nondecreasing rearrangement, i.e., there exists a nondecreasing sequence (b_{n}) such that (a_{n}) ∼ (b_{n}).

4 L-folds of weighted r-generalized Fibonacci powers

In this section, for each x ∈ ℝ, we let ⌈x⌉ denote the least integer greater than or equal to x. A well-known and widely used criterion for complete sequences is the following result given by Brown [1].

Theorem 10. [1, Theorem 1] Let (a_{n}) be a sequence of positive integers. If a_{1} = 1 and ∑_{k=1}^{n} a_{k} ≥ a_{n+1} − 1 for all n ∈ ℕ, then (a_{n}) is complete. The converse holds for increasing sequences of positive integers.

By the above theorem, for any sequence of positive integers (a_{n}), we see that if a_{1} = 1, and L is a positive integer such that L ≥ ⌈sup_{n∈ℕ} \frac{a_{n+1}−1}{∑_{k=1}^{n} a_{k}} ⌉, then the L-fold sequence of (a_{n}) is complete. Moreover, the converse holds when (a_{n}) is increasing. Then, for each sequence (a_{n}) of positive integers, we define L_{(a_{n})} := ⌈sup_{n∈ℕ} \frac{a_{n+1}−1}{∑_{k=1}^{n} a_{k}} ⌉. For an increasable sequence

(a_{n}) of positive integers, we also let L_{[a_{n}]} := ⌈sup_{n∈ℕ} \frac{b_{n+1}−1}{∑_{k=1}^{n} b_{k}} ⌉, where (b_{n}) is the nondecreasing sequence equivalent to (a_{n}).

Hunsucker and Wardlaw [6] showed that L_{[F_{n}^{m}]} = L_{(F_{n}^{m})} = 2^{m−1}, where (F_{n}^{m}) is the usual Fibonacci mth power. Computing L_{[a_{n}^{m}]} for an increasable weighted r-generalized Fibonacci mth power is not simple in general. So, in this section, we provide bounds for this value and give its exact value depending on finitely many terms of the sequence in some special cases.

The next lemma provides the least upper bound for the consecutive ratios of a weighted r-generalized Fibonacci sequence. Notice that the supremum depends on only the first r terms of the ratios, which is crucial information to obtain the upper bound for L_{[a_{n}^{m}]} in the following theorem.

Lemma 11. Let (a_{n}) be a weighted r-generalized Fibonacci sequence. Then

sup_{n∈ℕ} \frac{a_{n+1}}{a_{n}} = max_{1≤n≤r} \frac{a_{n+1}}{a_{n}} .

Proof. Let α := max_{1≤n≤r} \frac{a_{n+1}}{a_{n}}. Since

α − \frac{a_{n+1}}{a_{n}} = \frac{1}{a_{n}} ∑_{k=1}^{r} A_{k}a_{n−k} (α − \frac{a_{n−k+1}}{a_{n−k}} )

for all n > r and α − \frac{a_{k+1}}{a_{k}} ≥ 0 for all k ≤ r, we have that α − \frac{a_{n+1}}{a_{n}} ≥ 0 for all n ∈ ℕ by induction.

Theorem 12. Let (a_{n}) be an increasable weighted r-generalized Fibonacci sequence with some initial value equal to 1. Let k be a nonnegative integer and let b_{1} ≤ b_{2} ≤ … ≤ b_{k+1} be a nondecreasing rearrangement of a_{1}, a_{2}, …, a_{k+1}. Then, for all m ∈ ℕ,

L_{[a_{n}^{m}]} ≤ max(S_{k} ∪ T_{k}),

where S_{k} := \{⌈\frac{b_{i+1}^{m}−1}{∑_{l=1}^{i} b_{l}^{m}} ⌉ : 1 ≤ i ≤ k\} and T_{k} := \{⌈\frac{a_{k+i+1}^{m}}{a_{k+i}^{m}} ⌉ − 1 : 1 ≤ i ≤ r\}.

Proof. The theorem is obvious when a_{n} = 1 for all n ∈ ℕ. Then assume that (a_{n}) is not a constant sequence. Fix m ∈ ℕ, and let ρ : ℕ → ℕ be a bijection such that (a_{ρ(n)}) is nondecreasing. We show that

\frac{a_{ρ(n+1)}^{m} − 1}{∑_{i=1}^{n} a_{ρ(i)}^{m}} ≤ max(S_{k} ∪ T_{k})

by induction on n ∈ ℕ. Notice that if ρ(1) < k + 1, then a_{ρ(1)} = b_{1} and a_{ρ(2)} ≤ b_{2}. So

\frac{a_{ρ(2)}^{m} − 1}{a_{ρ(1)}^{m}} ≤ \frac{b_{2}^{m} − 1}{b_{1}^{m}} ≤ max S_{k}.

If ρ(1) ≥ k + 1, then by Lemma 11,

\frac{a_{ρ(2)}^{m} − 1}{a_{ρ(1)}^{m}} = \frac{a_{ρ(2)}^{m}}{a_{ρ(1)}^{m}} − 1 ≤ \frac{a_{ρ(1)+l}^{m}}{a_{ρ(1)+l−1}^{m}} − 1 ≤ max T_{k},

where l is the least positive integer such that a_{ρ(1)+l} > a_{ρ(1)}. Now, let n ∈ ℕ be such that \frac{a_{ρ(n+1)}^{m}−1}{∑_{i=1}^{n} a_{ρ(i)}^{m}} ≤ max(S_{k} ∪ T_{k}).

Case 1: Assume that there is no i > k such that a_{ρ(n+1)} = a_{i} and a_{ρ(n+2)} = a_{ρ(n+1)}. Let i′ ≤ k be minimal such that a_{ρ(n+1)} = b_{i′}. Then a_{ρ(n+2)}^{m} ≤ b_{i′+1}^{m} and ∑_{i=1}^{n+1} a_{ρ(i)}^{m} ≥ ∑_{i=1}^{i′} b_{i}^{m}. Thus,

\frac{a_{ρ(n+2)}^{m} − 1}{∑_{i=1}^{n+1} a_{ρ(i)}^{m}} ≤ \frac{b_{i′+1}^{m} − 1}{∑_{i=1}^{i′} b_{i}^{m}} ≤ max S_{k}.

Case 2: Assume that there is no i > k such that a_{ρ(n+1)} = a_{i} and a_{ρ(n+2)} > a_{ρ(n+1)}. Let i′ ≤ k be maximal such that a_{ρ(n+1)} = b_{i′}. Then a_{ρ(n+2)}^{m} ≤ b_{i′+1}^{m} and ∑_{i=1}^{n+1} a_{ρ(i)}^{m} ≥ ∑_{i=1}^{i′} b_{i}^{m}. Thus,

\frac{a_{ρ(n+2)}^{m} − 1}{∑_{i=1}^{n+1} a_{ρ(i)}^{m}} ≤ \frac{b_{i′+1}^{m} − 1}{∑_{i=1}^{i′} b_{i}^{m}} ≤ max S_{k}.

Case 3: Assume that there exists i > k such that a_{ρ(n+1)} = a_{i}. Then, by Lemma 11,

\frac{a_{ρ(n+2)}^{m} − a_{ρ(n+1)}^{m}}{a_{ρ(n+1)}^{m}} = \frac{a_{ρ(n+2)}^{m}}{a_{ρ(n+1)}^{m}} − 1 ≤ \frac{a_{ρ(n+1)+l}^{m}}{a_{ρ(n+1)+l−1}^{m}} − 1 ≤ max T_{k},

where l is the least positive integer such that a_{ρ(n+1)+l} > a_{ρ(n+1)}. After combining this result with the inductive hypothesis, we get

\frac{a_{ρ(n+2)}^{m} − 1}{∑_{i=1}^{n+1} a_{ρ(i)}^{m}} = \frac{(a_{ρ(n+1)}^{m} − 1) + (a_{ρ(n+2)}^{m} − a_{ρ(n+1)}^{m})}{(∑_{i=1}^{n} a_{ρ(i)}^{m}) + a_{ρ(n+1)}^{m}} ≤ max(S_{k} ∪ T_{k}).

If we let k = 0 in the previous theorem, we will get a nicer upper bound for L_{[a_{n}^{m}]}.

Corollary 13. Let (a_{n}) be an increasable weighted r-generalized Fibonacci sequence with at least one initial value equal to 1. Then, for all m ∈ ℕ,

L_{[a_{n}^{m}]} ≤ max_{1≤i≤r} ⌈(\frac{a_{i+1}}{a_{i}} )^{m}⌉ − 1.

Theorem 12 can also be used to find the exact value of L_{[a_{n}^{m}]} as shown in the following corollary and examples.

Corollary 14. Let (a_{n}) be an increasable weighted r-generalized Fibonacci sequence with at least one initial value equal to 1. Assume that there exists a rearrangement b_{1} ≤ b_{2} ≤ … ≤ b_{k+1} of a_{1}, a_{2}, …, a_{k+1} for some k ∈ ℕ such that b_{k+1} ≤ a_{n} for all n ≥ k + 1, and

max \{⌈\frac{a_{k+i+1}^{m}}{a_{k+i}^{m}} ⌉ − 1 : 1 ≤ i ≤ r\} ≤ max \{⌈\frac{b_{i+1}^{m} − 1}{∑_{l=1}^{i} b_{i}^{m}} ⌉ : 1 ≤ i ≤ k\} .

Then, L_{[a_{n}^{m}]} = max \{⌈\frac{b_{i+1}^{m}−1}{∑_{l=1}^{i} b_{i}^{m}} ⌉ : 1 ≤ i ≤ k\}.

Example 15. Let (a_{n}) be a generalized Fibonacci sequence with r = 2, A_{1} = A_{2} = 1, and (c_{1}, c_{2}) = (1, c) for some c ≥ 2. Then

L_{[a_{n}^{m}]} = ⌈\frac{a_{2}^{m} − 1}{∑_{i=1}^{1} a_{i}^{m}} ⌉ = c^{m} − 1

for all m ∈ ℕ.

Example 16. Let (a_{n}) be a weighted r-generalized Fibonacci sequence with r = 2, A_{1} = A_{2} = 1, and (c_{1}, c_{2}) = (c, 1) for some c ≥ 2. If (b_{n}) is a rearrangement of (a_{n}) that is increasing, then

L_{[a_{n}^{m}]} = ⌈\frac{b_{2}^{m} − 1}{∑_{i=1}^{1} b_{i}^{m}} ⌉ = c^{m} − 1

for all m ∈ ℕ. In particular, if c = 2, then (a_{n}) is the Lucas sequence.

Example 17. Let (a_{n}) be a weighted r-generalized Fibonacci sequence with r ≥ 2, A_{i} = P ∈ ℕ, and c_{i} = 1 for all i ≤ r. Then

L_{[a_{n}^{m}]} = ⌈\frac{a_{r+1}^{m} − 1}{∑_{i=1}^{r} a_{i}^{m}} ⌉ = P^{m}r^{m−1}

for all m ∈ ℕ. In particular, if r = 2 and P = 1, then this sequence is the usual Fibonacci sequence.

Proof. Notice that a_{i} = 1 for all i ≤ r and a_{r+j} = (P r−1)(P +1)^{j−1}+1 for j = 1, 2, …, r+1. We apply Corollary 14 with k = r. Since

\frac{a_{i+1}}{a_{i}} ≤ \frac{a_{j+1}}{a_{j}}

for r + 1 ≤ i ≤ j ≤ 2r, it suffices to show that

⌈\frac{a_{2r+1}^{m}}{a_{2r}^{m}} ⌉ − 1 ≤ \frac{a_{r+1}^{m}}{∑_{i=1}^{r} a_{i}^{m}} ,

which is equivalent to

(P + 1 − \frac{P}{(P r − 1)(P + 1)^{r−1} + 1})^{m} − 1 ≤ P^{m}r^{m−1},

for all m ∈ ℕ. If m = 1, then

P + 1 − \frac{P}{(P r − 1)(P + 1)^{r−1} + 1} − 1 ≤ P,

and we are done. If m ≥ 2 and P ≥ 2, then it is easy to see that (P + 1)^{m} − 1 ≤ P^{m}r^{m−1} and we are done. If m ≥ 2, P = 1 and r ≥ 3, then

(2 − \frac{1}{2^{r−1}(r − 1) + 1})^{m} < 2^{m} ≤ 3^{m−1} + 1 ≤ r^{m−1} + 1.

Lastly, if m ≥ 2, P = 1 and r = 2, then (\frac{5}{3})^{m} ≤ 2^{m−1} + 1 by induction on m.

Example 18. Let (a_{n}) be a weighted r-generalized Fibonacci sequence with r = 2, A_{1} = P, A_{2} = Q, and (c_{1}, c_{2}) = (1, P) for some P, Q ∈ ℕ. Then

L_{[a_{n}^{m}]} = ⌈\frac{a_{3}^{m} − 1}{∑_{i=1}^{2} a_{i}^{m}} ⌉ = ⌈\frac{(P^{2} + Q)^{m} − 1}{P^{m} + 1}⌉

for all m ∈ ℕ. This sequence is called (P, Q)-Fibonacci sequence [9]. In particular, if Q = 1, then this sequence is called Fibonacci P -sequence [3].

Proof. Note that in this case r = 2, and a_{1}, a_{2}, a_{3}, a_{4}, a_{5} are

1, P, P^{2} + Q, P^{3} + 2P Q, P^{4} + 3P^{2}Q + Q^{2},

respectively. First, fix m ≥ 2 and apply Corollary 14 with k = 2. Notice that

⌈\frac{a_{4}^{m}}{a_{3}^{m}} ⌉ − 1 = ⌈(\frac{P^{3} + 2P Q}{P^{2} + Q})^{m}⌉ − 1 ≤ ⌈(\frac{P^{4} + 3P^{2}Q + Q^{2}}{P^{3} + 2P Q} )^{m}⌉ − 1 = ⌈\frac{a_{5}^{m}}{a_{4}^{m}} ⌉ − 1

because (P^{3} + 2P Q)^{2} ≤ (P^{4} + 3P^{2}Q + Q^{2})(P^{2} + Q) by comparing coefficients. Also,

⌈\frac{a_{2}^{m} − 1}{∑_{i=1}^{1} a_{i}^{m}} ⌉ = P^{m} − 1 ≤ ⌈\frac{(P^{2} + Q)^{m} − 1}{P^{m} + 1}⌉ = ⌈\frac{a_{3}^{m} − 1}{∑_{i=1}^{2} a_{i}^{m}} ⌉

because (P^{m} − 1)(P^{m} + 1) ≤ (P^{2} + Q)^{m} − 1. Then it is left to show that

⌈\frac{a_{5}^{m}}{a_{4}^{m}} ⌉ − 1 ≤ ⌈\frac{a_{3}^{m} − 1}{∑_{i=1}^{2} a_{i}^{m}} ⌉ ,

which is equivalent to

⌈(\frac{P^{4} + 3P^{2}Q + Q^{2}}{P^{3} + 2P Q} )^{m}⌉ ≤ ⌈\frac{(P^{2} + Q)^{m} + P^{m}}{P^{m} + 1} ⌉ .

Case m ≥ 4: Since (P^{4} + 3P^{2}Q + Q^{2})^{4}(P^{4} + 1) ≤ (P^{2} + Q)^{4}(P^{3} + 2P Q)^{4} by comparing coefficients and P^{m} + 1 ≤ (P^{4} + 1)^{\frac{m}{4}} , we have that

(\frac{P^{4} + 3P^{2}Q + Q^{2}}{P^{3} + 2P Q} )^{m} ≤ \frac{(P^{2} + Q)^{m}}{P^{m} + 1} ≤ \frac{(P^{2} + Q)^{m} + P^{m}}{P^{m} + 1} .

Case m = 3: If Q ≥ 2, then (P^{4} + 3P^{2}Q + Q^{2})^{3}(P^{3} + 1) ≤ ((P^{2} + Q)^{3} + P^{3})(P^{3} + 2P Q)^{3} by comparing coefficients and we are done. Now, assume that Q = 1. Then

⌈\frac{(P^{2} + 1)^{3} + P^{3}}{P^{3} + 1} ⌉ = ⌈P^{3} + 3P + \frac{3P^{2} − 3P + 1}{P^{3} + 1}⌉ = P^{3} + 3P + 1.

Since (P^{4} + 3P^{2} + 1)^{3} ≤ (P^{3} + 3P + 1)(P^{3} + 2P)^{3} by comparing coefficients, we are done.

Case m = 2: If Q ≥ 2, then (P^{4} + 3P^{2}Q + Q^{2})^{2}(P^{2} + 1) ≤ ((P^{2} + Q)^{2} + P^{2})(P^{3} + 2P Q)^{2} by comparing coefficients and we are done. Now, assume that Q = 1. Then

⌈\frac{(P^{2} + 1)^{2} + P^{2}}{P^{2} + 1} ⌉ = ⌈P^{2} + 1 + \frac{P^{2}}{P^{2} + 1}⌉ = P^{2} + 2.

Since (P^{4} + 3P^{2} + 1)^{2} ≤ (P^{2} + 2)(P^{3} + 2P)^{2} by comparing coefficients, we are done.

Lastly, assume that m = 1. If P ≥ 2 and Q ≥ P + 2, then (P^{4} + 3P^{2}Q + Q^{2})(P + 1) ≤ (P^{2} + Q + P)(P^{3} + 2P Q) by comparing coefficients, so we are done in this subcase.

If P ∈ ℕ and Q ≤ P, then

\frac{P^{4} + 3P^{2}Q + Q^{2}}{P^{3} + 2P Q} ≤ P + 1 = ⌈\frac{P^{2} + Q + P}{P + 1} ⌉ .

For the last two subcases, we cannot use the same method as in the previous cases to obtain the result since the assumption in the Corollary 14 with k = 2 does not hold. Instead, if P ∈ ℕ and Q = P + 1, then one can show by induction that

⌈\frac{a_{3} − 1}{∑_{i=1}^{2} a_{i}} ⌉ = P = \begin{cases}\frac{a_{n+1}−1}{∑_{i=1}^{n} a_{n}} , & if n is even; \\ \frac{a_{n+1}}{∑_{i=1}^{n} a_{n}} > \frac{a_{n+1}−1}{∑_{i=1}^{n} a_{n}} , & if n is odd, \end{cases}

and we are done in this subcase.

If P = 1 and Q ≥ 3, then we apply Corollary 14 with k = 4. Notice that in this subcase, a_{1}, a_{2}, …, a_{7} are

1, 1, Q + 1, 2Q + 1, Q^{2} + 3Q + 1, 3Q^{2} + 4Q + 1, Q^{3} + 6Q^{2} + 5Q + 1,

respectively. Then

⌈\frac{a_{6}}{a_{5}} ⌉ − 1 = ⌈\frac{3Q^{2} + 4Q + 1}{Q^{2} + 3Q + 1}⌉ − 1 ≤ ⌈\frac{Q^{3} + 6Q^{2} + 5Q + 1}{3Q^{2} + 4Q + 1}⌉ − 1 = ⌈\frac{a_{7}}{a_{6}} ⌉ − 1.

Also,

⌈\frac{(2Q + 1) − 1}{1 + 1 + (Q + 1)}⌉ ≤ ⌈\frac{(Q^{2} + 3Q + 1) − 1}{1 + 1 + (Q + 1) + (2Q + 1)}⌉ ≤ ⌈\frac{(Q + 1) − 1}{1 + 1}⌉ ,

that is,

⌈\frac{a_{4} − 1}{∑_{i=1}^{3} a_{i}} ⌉ ≤ ⌈\frac{a_{5} − 1}{∑_{i=1}^{4} a_{i}} ⌉ ≤ ⌈\frac{a_{3} − 1}{∑_{i=1}^{2} a_{i}} ⌉ .

Since \frac{a_{7}}{a_{6}} − 1 = \frac{Q^{3}+6Q^{2}+5Q+1}{3Q^{2}+4Q+1} − 1 ≤ \frac{(Q+1)−1}{1+1} = \frac{a_{3}−1}{∑_{i=1}^{2} a_{i}}, we are done.

From the previous example, we see that it is difficult to obtain L_{[a_{n}^{m}]} when m is small. However, for a large value of m, we have the following result.

Theorem 19. Let (a_{n}) be a weighted r-generalized Fibonacci sequence. Assume that there is only one k_{0} ∈ \{1, 2, …, 2r\} such that \frac{a_{k_{0}+1}}{a_{k_{0}}} = max_{1≤k≤2r} \frac{a_{k+1}}{a_{k}}. If a_{k} ≤ a_{k_{0}} for all k ≤ k_{0}, then there exists M ∈ ℕ such that

L_{(a_{n}^{m})} = ⌈\frac{a_{k_{0}+1}^{m} − 1}{∑_{k=1}^{k_{0}} a_{k}^{m}}⌉

for all m ≥ M.

Proof. Since a_{k_{0}+1} > 1 and a_{i} ≤ a_{k_{0}} for all i ≤ k_{0}, there exists δ > 0 such that for all m ∈ ℕ,

\frac{a_{k_{0}+1}^{m} − 1}{∑_{i=1}^{k_{0}} a_{i}^{m}} = (\frac{1 − _{\frac{1}{a_{k_{0}+1}^{m}}}}{∑{k_{0} \atop i=1} \frac{a_{i}^{m}}{a_{k_{0}}^{m}}} ) \frac{a_{k_{0}+1}^{m}}{a_{k_{0}}^{m}} ≥ δ · \frac{a_{k_{0}+1}^{m}}{a_{k_{0}}^{m}} .

Let k′ ∈ ℕ be such that \frac{a_{k′+1}}{a_{k′}} = max\begin{matrix}1≤k≤2r \\ k≠k_{0} \end{matrix} \frac{a_{k+1}}{a_{k}}. Then, by Lemma 11, for all m, k ∈ ℕ with

k ≠ k_{0},

\frac{a_{k′+1}^{m}}{a_{k′}^{m}} ≥ \frac{a_{k+1}^{m}}{a_{k}^{m}} ≥ \frac{a_{k+1}^{m} − 1}{∑_{i=1}^{k} a_{i}^{m}}.

Since \frac{a_{k_{0}+1}}{a_{k_{0}}} > \frac{a_{k′+1}}{a_{k′}}, there exists M ∈ ℕ such that, for each m ≥ M,

\frac{a_{k_{0}+1}^{m} − 1}{∑_{i=1}^{k_{0}} a_{i}^{m}} ≥ δ · \frac{a_{k_{0}+1}^{m}}{a_{k_{0}}^{m}} ≥ \frac{a_{k′+1}^{m}}{a_{k′}^{m}} ≥ \frac{a_{k+1}^{m} − 1}{∑_{i=1}^{k} a_{i}^{m}}

for all k ∈ ℕ.

We close this section by providing a lower bound for L_{[a_{n}^{m}]} related to the dominant root, when (a_{n}) is in a primitive form.

Theorem 20. Let (a_{n}) be a weighted r-generalized Fibonacci sequence in a primitive form, and λ the dominant root of its characteristic polynomial. Then L_{[a_{n}^{m}]} ≥ ⌈λ^{m}⌉ − 1 for all m ∈ ℕ.

Proof. The theorem is obvious when λ = 1. Now, assume that λ > 1 and fix m ∈ ℕ. By Proposition 3, there exists C > 0 such that ∑_{k=1}^{n} a_{k}^{m} = Cλ^{nm} + o(λ^{nm}) as n → ∞. Then

lim_{n→∞} \frac{a_{n+1}^{m} − 1}{∑_{k=1}^{n} a_{k}^{m}} + 1 = lim_{n→∞} \frac{Cλ^{(n+1)m} + o(λ^{nm})}{Cλ^{nm} + o(λ^{nm})} = λ^{m}.

5 Generalized distinguished sequences

Before we define the distinguished sequences, let us present some important tools to study a weighted r-generalized Fibonacci sequence that is not in a primitive form. The following result tells us that, for any weighted r-generalized Fibonacci sequence (a_{n}), the multiset {a_{n} : n ∈ ℕ} can be decomposed into a disjoint union of {b_{n}^{(i)} : n ∈ ℕ}’s, where (b_{n}^{(i)}) is a weighted r-generalized Fibonacci sequence in a primitive form for all i.

Proposition 21. Let (a_{n}) be a weighted r-generalized Fibonacci sequence. We also let d := gcd {k : A_{k} > 0}. For 1 ≤ i ≤ d, let b_{n}^{(i)} := a_{(n−1)d+i} for all n ∈ ℕ. Then the sequences (b_{n}^{(1)}), (b_{n}^{(2)}), …, (b_{n}^{(d)}) satisfy a common recurrence relation that is in a primitive form, and

[a_{n}] = [b_{n}^{(1)}] ⊕ [b_{n}^{(2)}] ⊕ ⋯ ⊕ [b_{n}^{(d)}].

Proof. Clearly, [a_{n}] = [b_{n}^{(1)}] ⊕ [b_{n}^{(2)}] ⊕ … ⊕ [b_{n}^{(d)}]. Notice that, for 1 ≤ i ≤ d and n > \frac{r}{d},

b_{n}^{(i)} = ∑_{k=1}^{\frac{r}{d}} A_{kd}b_{n−k}^{(i)},

and gcd {k : A_{kd} > 0} = 1. Hence, (b_{n}^{(i)}) is in a primitive form for all i.

By the previous result and Proposition 3, we have the following consequence.

Corollary 22. Let (a_{n}) be a weighted r-generalized Fibonacci sequence and d := gcd{k : A_{k} > 0}. We also let b_{n}^{(i)} be as in Proposition 21, for all 1 ≤ i ≤ d, and λ > 1 be the dominant root of the characteristic polynomial of (b_{n}^{(1)}). Then there exist C_{1}, C_{2}, …, C_{d} ∈ [1, λ) and k_{1}, k_{2}, …, k_{d} ∈ ℤ such that

a_{nd+i} = C_{i}λ^{n+k_{i}} + o(λ^{n}) for all i ∈ {1, 2, …, d}, as n → ∞.

From the previous section, we have obtained complete sequences of weighted r-generalized Fibonacci mth powers. However, these sequences may not be minimal. Then, in this section, we will study a particular minimal sequence introduced by Hunsucker and Wardlaw in [6], called a distinguished sequence, of a weighted r-generalized Fibonacci power.

Definition 23. A distinguished sequence of a sequence of positive integers (s_{n}) is a nondecreasing complete sequence (d_{n}) that maps ℕ onto {s_{n} : n ∈ ℕ}, and for all n ∈ ℕ, 1 + ∑_{k=1}^{n−1} d_{k} < d_{n+1} whenever d_{n} < d_{n+1}.

Although the distinguished sequence of Fibonacci mth power uniquely exists for each m ∈ ℕ [6, Section 4], the distinguished sequence of mth power of a weighted r-generalized Fibonacci (a_{n}^{m}) may not exist for all m ∈ ℕ.

Example 24. Let (a_{n}) be such that a_{1} = 1, a_{2} = 2, a_{3} = 3, a_{4} = 4 and a_{n} = a_{n−4} for all n ≥ 5. Then the distinguished sequence of (a_{n}) does not exist since 1 + a_{1} + a_{2} = a_{4}.

Although the distinguished sequence of a weighted r-generalized Fibonacci power may not exist in general, the distinguished sequence of its mth power uniquely exists when the weighted r-generalized Fibonacci sequence is in a primitive form and m is large enough.

Lemma 25. Let (s_{n}) be a strictly increasing sequence of positive integers with s_{1} = 1 such that s_{n} + s_{n+1} ≤ s_{n+2} for all n ∈ ℕ. Then there exists a unique distinguished sequence of (s_{n}).

Proof. Construct a distinguished sequence (d_{n}) from (s_{n}) as follows: For each i ∈ ℕ, start from i = 1, add multiple s_{i}’s to the sequence (d_{n}) so that

(1) (d_{n}) is nondecreasing,

(2) 1 + ∑_{\begin{matrix}k∈ℕ \\ d_{k}≤s_{i} \end{matrix}} d_{k} ≥ s_{i+1}, and

(3) 1 − s_{i} + ∑_{\begin{matrix}k∈ℕ \\ d_{k}≤s_{i} \end{matrix}} d_{k} < s_{i+1}.

Since s_{n} + s_{n+1} ≤ s_{n+2} for all n ∈ ℕ, the sequence (d_{n}) can be always constructed so that condition (3) is satisfied. Moreover, the uniqueness of the distinguished sequence results from the properties (2) and (3) of (d_{n}).

Theorem 26. Let (a_{n}) be an increasable weighted r-generalized Fibonacci sequence in a primitive form with at least one initial value equal to 1. Then there exists M ∈ ℕ such that the distinguished sequence of (a_{n}^{m}) uniquely exists for all m ≥ M.

Proof. Let λ ≥ 1 be the dominant root of the characteristic polynomial of (a_{n}). If λ = 1, then we are done because a_{n} = 1 for all n ∈ ℕ. Now, assume that λ > 1. Let (b_{n}) be a rearrangement of (a_{n}) such that (b_{n}) is nondecreasing. Since a_{n} = Cλ^{n} + o(λ^{n}) for some C > 0 and all large n ∈ ℕ, there exists N ∈ ℕ such that a_{n} = b_{n} for all n ≥ N. Hence, there exists M ∈ ℕ such that, for each n ∈ ℕ, if b_{n+2} > b_{n+1}, then

b_{n}^{m} + b_{n+1}^{m} ≤ b_{n+2}^{m} for all m ≥ M

because λ > 1 and lim_{n→∞} \frac{b_{n}^{m}+b_{n+1}^{m}}{b_{n+2}^{m}} = (\frac{1}{λ^{2m}} + \frac{1}{λ^{m}} ) for all m ∈ ℕ. Let (s_{n}) be a strictly increasing sequence that maps onto {b_{n}^{m} : n ∈ ℕ}. By Lemma 25, we obtain a unique distinguished sequence of (s_{n}) and we are done.

The previous theorem does not hold for weighted r-generalized Fibonacci sequences that are not in a primitive form in general.

Example 27. Let a_{1} = 1, a_{2} = 2, a_{3} = 3, a_{4} = 4, a_{5} = 5, a_{6} = 4, a_{7} = 3, a_{8} = 2, and a_{n} = a_{n−4} + 2a_{n−8} for all n > 8. Then, for all n ∈ ℕ,

a_{4(n−1)+i} = \begin{cases}2^{n} + (−1)^{n}, & if i = 1; \\ 2^{n}, & if i = 2; \\ 2^{n} − (−1)^{n}, & if i = 3; \\ 2^{n} − 2(−1)^{n}, & if i = 4. \end{cases}

Let m ∈ ℕ. Notice that in any nondecreasing rearrangement (b_{n}) of (a_{n}), there are infinitely many four consecutive terms b_{l} < b_{l+1} < b_{l+2} < b_{l+3} in the form of a_{4(2n−1)+4} < a_{4(2n−1)+3} < a_{4(2n−1)+2} < a_{4(2n−1)+1}, for some n ∈ ℕ, such that

b_{l}^{m} + b_{l+1}^{m} = (2^{2n} − 2)^{m} + (2^{2n} − 1)^{m} ≥ (2^{2n} + 1)^{m} = b_{l+3}^{m}.

Hence, there is no distinguished sequence of (a_{n}^{m}).

The next theorem provides information on the repetition of a_{n}^{m} in its distinguished sequence for large m, n ∈ ℕ.

Theorem 28. Let (a_{n}) be an increasable weighted r-generalized Fibonacci sequence in a primitive form, M be as in the previous theorem, m ≥ M, and (d_{n}) be the distinguished sequence of (a_{n}^{m}). We also let s be the greatest integer less than or equal to λ^{m}, where λ := lim_{n→∞} \frac{a_{n+1}}{a_{n}}. If λ > 1, then there exists N such that, for each n ≥ N, d_{k} = a_{n}^{m} for exactly s or s − 1 values of k. In particular, if λ = 1, then d_{n} = 1 for all n ∈ ℕ.

Proof. Assume that λ > 1. Then, by the definition of limits, there exists N ∈ ℕ such that s − \frac{λ−1}{2λ} < (\frac{a_{n+1}}{a_{n}} )^{m} < s + 1 and (\frac{a_{n−1}}{a_{n}} )^{m} < \frac{1}{λ} + \frac{λ−1}{2λ} for all n ≥ N. Fix n ≥ N and let n_{0} := min{k : d_{k} = a_{n}^{m}}. We show that

1 + ∑_{k=1}^{n_{0}−1} d_{k} + (s − 2)a_{n}^{m} < a_{n+1}^{m} ≤ 1 + ∑_{k=1}^{n_{0}−1} d_{k} + sa_{n}^{m}.

Since (d_{n}) is a distinguished sequence,

1 + ∑_{k=1}^{n_{0}−1} d_{k} < a_{n−1}^{m} + a_{n}^{m}.

So

1 + ∑_{k=1}^{n_{0}−1} d_{k} + (s − 2)a_{n}^{m} < a_{n−1}^{m} + (s − 1)a_{n}^{m} < (s − \frac{λ − 1}{2λ}) a_{n}^{m} < a_{n+1}^{m}.

Also, notice that

a_{n+1}^{m} < (1 + s)a_{n}^{m} ≤ 1 + ∑_{k=1}^{n_{0}−1} d_{k} + sa_{n}^{m}.

In general, Theorem 26 does not hold for weighted r-generalized Fibonacci sequences that are not in a primitive form. However, it does hold for weighted r-generalized Fibonacci sequences satisfying some particular conditions as shown in the following theorem and example.

Theorem 29. Let (a_{n}) be an increasable weighted r-generalized Fibonacci sequence with a_{i} = 1 for some i ≤ r. Assume that [a_{n}] = [b_{n}^{(1)}] ⊕ [b_{n}^{(2)}] ⊕ … ⊕ [b_{n}^{(d)}], where d ≥ 2 and (b_{n}^{(k)}) is as in Proposition 21 for all k ≤ d. Let λ > 1 be the dominant root of the characteristic polynomial of (b_{n}^{(1)}). If lim_{n→∞} log_{λ} (\frac{b_{n}^{(p)}}{b_{n}^{(q)}} ) ∉ ℤ for all 1 ≤ p < q ≤ d, then for all but finitely many m ∈ ℕ, there exists a unique distinguished sequence of (a_{n}^{m}).

Proof. Assume that lim_{n→∞} log_{λ} (\frac{b_{n}^{(p)}}{b_{n}^{(q)}} ) ∉ ℤ for all 1 ≤ p < q ≤ s. Then, by Corollary 22, there exist k_{1}, k_{2}, …, k_{d} ∈ ℤ and distinct numbers C_{1}, C_{2}, …, C_{d} ∈ [1, λ) such that

a_{nd+i} = C_{i}λ^{n+k_{i}} + o(λ^{n}) for all i ∈ {1, 2, …, d}, as n → ∞.

Without loss of generality, we may assume that 1 ≤ C_{1} < C_{2} < … < C_{d} < λ. Let (b_{n}) be a nondecreasing rearrangement of (a_{n}). Then there exists k ∈ ℤ such that, for each i ∈ {1, 2, …, d}, we have that b_{nd+i} = C_{i}λ^{n+k} +o(λ^{n}) as n → ∞. Since 1 ≤ C_{1} < C_{2} < … < C_{d} < λ, there exist α < 1 and N ∈ ℕ such that \frac{b_{n}}{b_{n+1}} ≤ α for all n ≥ N. Thus, there exists M ∈ ℕ such that whenever b_{n+2} > b_{n+1} for n ∈ ℕ and b_{n}^{m} + b_{n+1}^{m} ≤ b_{n+2}^{m} for all m ≥ M. Let (s_{n}) be a strictly increasing sequence that maps onto {b_{n}^{m} : n ∈ ℕ}. By Lemma 25, we obtain a unique distinguished sequence of (s_{n}) and we are done.

Example 30. Let a_{1} = 2, a_{2} = a_{3} = a_{4} = 1, and a_{n} = a_{n−2} + a_{n−4} for all n > 4. Then [a_{n}] = [L_{n}] ⊕ [F_{n}], where (L_{n}) is the sequence of Lucas numbers and (F_{n}) is the Fibonacci sequence. Notice that the dominant root of the characteristic polynomial of (L_{n}) is φ, where φ = \frac{1+\sqrt{5}}{2} is the golden ratio. It is also well-known that, for all n ∈ ℕ,

L_{n} = φ^{n−1} + (1 − φ)^{n−1} and F_{n} = \frac{φ^{n} − (1 − φ)^{n}}{\sqrt{5}} .

Thus

lim_{n→∞} log_{φ} (\frac{L_{n}}{F_{n}} ) = log_{φ}(\sqrt{5}) − 1 ∉ ℤ.

By Theorem 29, there exists a unique distinguished sequence of (a_{n}^{m}) for all but finitely many m ∈ ℕ.

The converse of Theorem 29 does not hold in general. For instance, it does not hold for any sequence that can be decomposed into multiple identical sequences as shown in the next example.

Example 31. Let a_{1} = 1, a_{2} = 1, and a_{n} = 2a_{n−2} for all n > 2. Then [a_{n}] = [b_{n}^{(1)}] ⊕ [b_{n}^{(2)}], where b_{1}^{(i)} = 1 and b_{n}^{(i)} = 2b_{n−1}^{(i)}, for all i ∈ {1, 2} and n > 1. Notice that the dominant root of the characteristic polynomial of (b_{n}^{(1)}) is λ = 2. Moreover,

lim_{n→∞} log_{2} (\frac{b_{n}^{(1)}}{b_{n}^{(2)}} ) = 0 ∈ ℤ.

However, since {a_{n} : n ∈ ℕ} = {b_{n}^{(1)} : n ∈ ℕ}, by using Theorem 26 with the sequence (b_{n}^{(1)}), there exists a unique distinguished sequence of (a_{n}^{m}) for all but finitely many m ∈ ℕ.

As we see in Example 27, it may happen that the distinguished sequence of the mth powers of a weighted r-generalized Fibonacci does not exist for any m ∈ ℕ. However, if we generalize Definition 23 as in the following definition, then this generalized distinguished sequence of the weighted r-generalized Fibonacci mth power exists for all but finitely many m ∈ ℕ.

Definition 32. A generalized distinguished sequence of a sequence of positive integers (s_{n}) is a nondecreasing complete sequence (d_{n}) that is minimal and maps ℕ onto {s_{n} : n ∈ ℕ}.

Theorem 33. Let (a_{n}) be an increasable weighted r-generalized Fibonacci sequence with at least one initial value equal to 1. If (a_{n}) is not a constant sequence, then there exists a generalized distinguished sequence of (a_{n}^{m}) for all large m ∈ ℕ.

Proof. Assume that (a_{n}) is not a constant sequence. Then, by Corollary 22, there exist λ > 1, C_{1}, C_{2}, …, C_{d} ∈ [1, λ), and k_{1}, k_{2}, …, k_{d} ∈ ℤ such that

a_{nd+i} = C_{i}λ^{n+k_{i}} + o(λ^{n}) for all i ∈ {1, 2, …, d}, as n → ∞.

Without loss of generality, we may assume that 1 ≤ C_{1} ≤ C_{2} ≤ … ≤ C_{d} < λ. Let (b_{n}) be a nondecreasing rearrangement of (a_{n}). Then there exists k ∈ ℤ such that, for all i ∈ \{1, 2, …, d\}, we have b_{nd+i} = C_{i}λ^{n+k} + o(λ^{n}) as n → ∞. Since \frac{C_{d}}{C_{1}λ} < 1 and

lim_{n→∞} \frac{(d + 1)b_{nd+d}^{m}}{b_{(n+1)d+1}^{m}} = (d + 1) (\frac{C_{d}}{C_{1}λ})^{m} for all m ∈ ℕ,

there exist M, N ∈ ℕ such that, for all m ≥ M

• b_{Nd}^{m} < b_{Nd+1}^{m},

• if n ≥ N, then (d + 1)b_{nd+d}^{m} ≤ b_{(n+1)d+1}^{m}, and

• if n ≤ Nd + 1 and b_{n−1} < b_{n}, then b_{n−2}^{m} + b_{n−1}^{m} ≤ b_{n}^{m}.

Now, fix m ≥ M. Let (s_{n}) be a strictly increasing finite sequence that maps onto {b_{n}^{m} : n ≤ Nd + 1}. Similarly to Lemma 25, we can construct a finite nondecreasing sequence d_{1}, d_{2}, …, d_{l} ∈ {b_{n}^{m} : n ≤ Nd + 1} such that • d_{l} = b_{Nd+1}^{m},

• {d_{1}, d_{2}, …, d_{l−1}} = {b_{n}^{m} : n ≤ Nd},

• 1 + ∑_{i=1}^{j} d_{i} ≥ d_{j+1} for all j < l, and

• 1 + ∑_{i=1}^{j−1} d_{i} < d_{j+1} for all j < l with d_{j} < d_{j+1}.

Next, let us append b_{Nd+1}^{m}, b_{Nd+2}^{m}, …, b_{(N+1)d+1}^{m} to our sequence (d_{n}). Notice that 1 + ∑_{i=1}^{l−1} d_{i} < d_{l} + d_{l−1} ≤ 2b_{Nd+d}^{m}. Thus,

b_{(N+1)d+1}^{m} − (1 + ∑_{i=1}^{l−1} d_{i} + ∑_{i=Nd+2}^{Nd+d} b_{i}^{m}) > b_{(N+1)d+1}^{m} − (d + 1)b_{Nd+d}^{m} ≥ 0.

Let p := ⌈\frac{1}{b_{Nd+1}^{m}} (b_{(N+1)d+1}^{m} − (1 + ∑_{i=1}^{l−1} d_{i} + ∑_{i=Nd+2}^{Nd+d} b_{i}^{m}))⌉ ∈ ℕ. Define d_{l+i} := b_{Nd+1}^{m} for all i ∈ {1, …, p − 1}, and d_{l+p+j−2} := b_{Nd+j}^{m} for all j ∈ {2, 3, …, d + 1}. Then

1 + ( ∑_{i=1}^{l+p+d−2} d_{i}) − d_{j} ≤ 1 + ( ∑_{i=1}^{l+p+d−2} d_{i}) − b_{Nd+1}^{m}

= (p − 1)b_{Nd+1}^{m} + 1 + ∑_{i=1}^{l−1} d_{i} + ∑_{i=Nd+2}^{Nd+d} b_{i}^{m}

< b_{(N+1)d+1}^{m}

= d_{l+p+d−1}

for l ≤ j < l + p + d − 1. For j ∈ {l, l + 1, …, l + p − 2}, since d_{j} = d_{j+1}, we have that

1 + ∑_{i=1}^{j} d_{i} ≥ d_{j+1}.

Moreover, for all j ∈ {l + p − 1, l + p, …, l + p + d − 3},

1 + ∑_{i=1}^{j} d_{i} = 1 + pb_{Nd+1}^{m} + ∑_{i=1}^{l−1} d_{i} + ∑_{i=Nd+2}^{Nd+j−l−p+2} b_{i}^{m}

≥ b_{(N+1)d+1}^{m} − ∑_{i=Nd+j−l−p+3}^{Nd+d} b_{i}^{m}

≥ b_{(N+1)d+1}^{m} − (d − 1)b_{Nd+d}^{m}

= b_{(N+1)d+1}^{m} − db_{Nd+d}^{m} + b_{Nd+d}^{m}

≥ b_{Nd+d}^{m} ≥ d_{j+1}, and

1 + ∑_{i=1}^{l+p+d−2} d_{i} = 1 + pb_{Nd+1}^{m} + ∑_{i=1}^{l−1} d_{i} + ∑_{i=Nd+2}^{Nd+d} b_{i}^{m} ≥ b_{(N+1)d+1}^{m} = d_{l+p+d−1}.

After this process, we have that b_{Nd+1}^{m}, b_{Nd+2}^{m}, …, b_{(N+1)d+1}^{m} ∈ {d_{n}}. After doing this process for b_{nd+1}^{m}, b_{nd+2}^{m}, …, b_{(n+1)d+1}^{m}, for all n ≥ N + 1, we obtain a generalized distinguished sequence (d_{n}) for (a_{n}^{m}).

We illustrate the proof of the previous theorem by the following example.

Example 34. Let (a_{n}) be as in Example 27 and let (b_{n}) be its nondecreasing rearrangement, which is

1, 2, 2, 3, 3, 4, 4, 5 , 7, 8, 9, 10 , 14, 15, 16, 17 , 31, 32, 33, 34 , ….

According to the proof, d = 4, λ = 2, and C_{1} = C_{2} = C_{3} = C_{4} = 1. Let N = 2 and M = 5. Notice that for all m ≥ 5,

• b_{8}^{m} < b_{9}^{m},

• if n ≥ 2, then 5 · b_{4n+4}^{m} ≤ b_{4(n+1)+1}^{m}, and

• if n ≤ 9 and b_{n−1} < b_{n}, then b_{n−2}^{m} + b_{n−1}^{m} ≤ b_{n}^{m}.

Now, fix m = 5. We show that there exists a generalized distinguished sequence of (a_{n}^{5}). As in the proof of Theorem 26, there is a finite nondecreasing sequence d_{1}, d_{2}, …, d_{50} ∈

{1^{5}, 2^{5}, 3^{5}, 4^{5}, 5^{5}, 7^{5}} with the number of 1^{5}, 2^{5}, 3^{5}, 4^{5}, 5^{5}, 7^{5} equal to 31, 7, 4, 2, 5, 1, respectively, such that

• 1 + ∑_{i=1}^{j} d_{i} ≥ d_{j+1} for all j < 50 and

• 1 + ∑_{i=1}^{j−1} d_{i} < d_{j+1} for all j < 50 with d_{j} < d_{j+1}.

Next, we add 7^{5}, 8^{5}, 9^{5}, 10^{5} to (d_{n}). In this case, according to the proof,

p = ⌈\frac{1}{7^{5}} (14^{5} − (1 + (31 · 1^{5} + 7 · 2^{5} + 4 · 3^{5} + 2 · 4^{5} + 5 · 5^{5}) + (8^{5} + 9^{5} + 10^{5})))⌉
= 20.

Then we define d_{51}, d_{52}, …, d_{69} to be 7^{5}, and define d_{70}, d_{71}, d_{72}, d_{73} to be 8^{5}, 9^{5}, 10^{5}, 14^{5} respectively.

If we repeat the process in the previous paragraph with each of the following 4 consecutive terms, we will obtain a generalized distinguished sequence for (a_{n}^{5}).

6 Acknowledgments

The authors wish to thank Department of Mathematics, Faculty of Science, Silpakorn University. We would also like to thank anonymous reviewers for valuable and constructive comments and suggestions.

References

[1] J. L. Brown, Jr., Note on complete sequences of integers, Amer. Math. Monthly 68 (1961), 557–560.

[2] F. Dubeau, W. Motta, M. Rachidi, and O. Saeki, On weighted r-generalized Fibonacci sequences, Fibonacci Quart. 35 (1997), 102–110.

[3] S. Falcón and Á. Plaza, On the Fibonacci k-numbers, Chaos Solitons Fractals 32 (2007), 1615–1624.

[4] A. Fiorenza and G. Vincenzi, Limit of ratio of consecutive terms for general order-k linear homogeneous recurrences with constant coefficients, Chaos Solitons Fractals 44 (2011), 145–152.

[5] V. E. Hoggatt, Jr. and C. King, Problem E1424, Amer. Math. Monthly 67 (1960), 593.

[6] J. L. Hunsucker and W. P. Wardlaw, Complete sequences of Fibonacci powers, Fibonacci Quart. 11 (1973), 387–394.

[7] R. O’Connell, Representations of integers as sums of Fibonacci squares, Fibonacci Quart. 10 (1972), 103–111.

[8] A. M. Ostrowski, Solution of Equations in Euclidean and Banach Spaces, Academic Press, 1973.

[9] A. Suvarnamani and M. Tatong, Some properties of (p, q)-Fibonacci numbers, Sci. & Tech. RMUTT J. 5 (2015), 17–21.

2020 Mathematics Subject Classification: Primary 11B39; Secondary 11B37. Keywords: weighted r-generalized Fibonacci sequence, complete sequence, Fibonacci power, generalized distinguished sequence.

(Concerned with sequences \underline{A000045}, \underline{A000032}, and \underline{A007598}.)

Received September 13 2020; revised versions received September 14 2020; February 10 2021; February 26 2021. Published in Journal of Integer Sequences, April 25 2021.

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