On generalized Leonardo p-numbers

Kantaphon Kuhapatanakul and Pongpol Ruankong Department of Mathematics Faculty of Science Kasetsart University Bangkok 10900 Thailand

                       fscikpkk@ku.ac.th
                       fscippru@ku.ac.th

Abstract We introduce a generalization of Leonardo p-numbers, which extends the definition of Leonardo p-numbers given by Tan and Leung, and investigate some of their basic properties. Moreover, we give a companion matrix and derive some identities regarding the numbers.

1 Introduction

The sequence of the Fibonacci numbers \underline{A000045} \{F_{n}\}_{n≥0} is the second-order linear recurrence sequence given by

F_{0} = 0, F_{1} = 1 and F_{n+1} = F_{n} + F_{n−1} for n ≥ 1.

One of the generalizations of Fibonacci numbers, given by Stakhov [7], is the Fibonacci p-numbers F_{p}(n), which is defined for positive integers p by

F_{p}(n) = F_{p}(n − 1) + F_{p}(n − p − 1) for n > p

with the initial values F_{p}(0) = 0 and F_{p}(1) = F_{p}(2) = ⋯ = F_{p}(p) = 1.

Furthermore, Catarino and Borges in [2] introduced the Leonardo sequence {Le_{n}}_{n≥0} and derived some of its properties. It is the sequence \underline{A001595} satisfying the recurrence relation

Le_{0} = Le_{1} = 1 and Le_{n} = Le_{n−1} + Le_{n−2} + 1 for n ≥ 2.

The first author and Chobsorn [4] presented the generalized Leonardo sequence and obtained some properties of this sequence. Recently, Tan and Leung [8] have introduced the sequence of the Leonardo p-numbers {L_{p,n}}_{n≥0} as

L_{p}(n) = L_{p}(n − 1) + L_{p}(n − p − 1) + p for n > p

with the initial values L_{p}(0) = L_{p}(1) = ⋯ = L_{p}(p) = 1. When p = 1, the Leonardo p-numbers reduce to the Leonardo numbers.

In this paper, we extend the aforementioned definition of the Leonardo p-numbers and derive some properties of the new sequence.

2 Generalized Leonardo p-numbers

We begin by defining a sequence of the generalized Leonardo p-numbers and establishing its basic properties.

Definition 1. For all fixed pairs of positive integers p and k, define the generalized Leonardo p-sequence {L_{p,k}(n)}_{n≥0} by

L_{p,k}(n) = L_{p,k}(n − 1) + L_{p,k}(n − p − 1) + k for n > p

with the initial values L_{p,k}(0) = L_{p,k}(1) = ⋯ = L_{p,k}(p) = 1.

Some special cases of the generalized Leonardo p-numbers are as follows.

• If k = p, the generalized Leonardo p-numbers are the Leonardo p-numbers.

• If p = 1, the generalized Leonardo p-numbers are the generalized Leonardo numbers.

• If p = 1 and k = 2, this sequence reduces to the sequence \underline{A111314}.

We compute the first few terms of the sequences F_{2}(n) and L_{2,k}(n) for k = 1, 2, 3, 4, 5.

\begin{matrix}n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 \\ F_{2}(n) & 1 & 1 & 1 & 2 & 3 & 4 & 6 & 9 & 13 & 19 & 28 & 41 & 60 & 88 \\ L_{2,1}(n) & 1 & 1 & 3 & 5 & 7 & 11 & 17 & 25 & 37 & 55 & 81 & 119 & 175 & 257 \\ L_{2,2}(n) & 1 & 1 & 4 & 7 & 10 & 16 & 25 & 37 & 55 & 82 & 121 & 178 & 262 & 385 \\ L_{2,3}(n) & 1 & 1 & 5 & 9 & 13 & 21 & 33 & 49 & 73 & 109 & 161 & 237 & 349 & 513 \\ L_{2,4}(n) & 1 & 1 & 6 & 11 & 16 & 26 & 41 & 61 & 91 & 136 & 201 & 296 & 436 & 641 \\ L_{2,5}(n) & 1 & 1 & 7 & 13 & 19 & 31 & 49 & 73 & 109 & 163 & 241 & 355 & 523 & 769 \end{matrix}

Table 1: The first 14 terms of F_{2}(n), L_{2,1}(n), L_{2,2}(n), L_{2,3}(n), L_{2,4}(n) and L_{2,5}(n).

The Fibonacci 2-numbers F_{2}(n) and the generalized Leonardo 2-numbers L_{2,2}(n) are the sequences \underline{A000930} and \underline{A362255}, respectively.

The non-homogeneous recurrence relation of the generalized Leonardo p-numbers can be rewritten as the following homogeneous recurrence relation:

L_{p,k}(n) = L_{p,k}(n − 1) + L_{p,k}(n − p) − L_{p,k}(n − 2p − 1) for n > 2p.

Next, we establish a connection between the generalized Leonardo p-numbers and the Fibonacci p-numbers.

Theorem 2. For positive integers p, k and a nonnegative integer n, we have

L_{p,k}(n) = (k + 1)F_{p}(n + 1) − k.  (1)

Proof. It is easy to check that the identity (1) holds for n = 0, 1, 2, …, p. Assume that, for some n ≥ p, the identity (1) holds for all integers m such that 0 ≤ m ≤ n. By the definition and the inductive hypothesis, we obtain

L_{p,k}(n + 1) = L_{p,k}(n) + L_{p,k}(n − p) + k
= (k + 1)F_{p}(n + 1) − k + (k + 1)F_{p}(n − p + 1) − k + k
= (k + 1)F_{p}(n + 2) − k,

which shows that the identity (1) holds for n + 1, thereby proving the theorem.

Theorem 3. For positive integers p, k and a nonnegative integer n, we have

∑_{i=0}^{n} L_{p,k}(i) = L_{p,k}(n + p + 1) − k(n + 1) − 1.

Proof. By the definition of L_{p,k}(n), write

L_{p,k}(i) = L_{p,k}(i + p + 1) − L_{p,k}(i + p) − k.

Summing a telescoping series, we get

∑_{i=0}^{n} L_{p,k}(i) = L_{p,k}(n + p + 1) − L_{p,k}(p) − k(n + 1).

Since L_{p,k}(p) = 1, we obtain the desired result.

Theorem 4. Let m and n be nonnegative integers. For positive integers p, k, we have

∑_{i=0}^{n} L_{p,k}((p + 1)i + m) = \begin{cases}L_{p,k} ((p + 1)n + m + 1) − kn, & if 0 ≤ m < p; \\ L_{p,k} ((p + 1)(n + 1)) − k(n + 1) − 1, & if m = p. \end{cases}

Proof. Since

L_{p,k}(i(p + 1) + m) = L_{p,k} (i(p + 1) + m + 1) − L_{p,k} ((i − 1)(p + 1) + m + 1) − k,

by summing a telescoping series we get

∑_{i=0}^{n} L_{p,k}(i(p + 1) + m) = L_{p,k} ((p + 1)n + m + 1) − L_{p,k}(−p + m) − k(n + 1).

Since L_{p,k}(−i) = −k for 1 ≤ i ≤ p and L_{p,k}(0) = 1, the result follows.

Here are some examples of Theorem 4 where p = 2.

(i) ∑_{i=0}^{n} L_{2,k}(3i) = L_{2,k} (3n + 1) − kn

(ii) ∑_{i=0}^{n} L_{2,k}(3i + 1) = L_{2,k} (3n + 2) − kn

(iii) ∑_{i=0}^{n} L_{2,k}(3i + 2) = L_{2,k} (3n + 3) − k(n + 1) − 1.

Moreover, substituting k = p in Theorem 2, Theorem 3 and Theorem 4, we obtain the identities of Theorem 1, Proposition 2 and Theorem 2, in [8], respectively.

3 Main results

In this section, we introduce a companion matrix of the generalized Leonardo p-numbers for p ≥ 2 (for the case p = 1, see [4, page 4]). For a given integer p ≥ 2, let H(p) consist of all those (2p + 1)^{th} order recurrence sequences {H_{p}(n)}_{n∈ℤ} of real numbers satisfying the relation

H_{p}(n) = H_{p}(n − 1) + H_{p}(n − p) − H_{p}(n − 2p − 1).

Note that the sequences L_{p,k}(n) are elements of H(p).

Two significant sequences A_{p} and B_{p} in H(p) are defined with the following initial values:

(i) A_{p}(1) = A_{p}(2) = ⋯ = A_{p}(p) = 1 and A_{p}(0) = A_{p}(−1) = ⋯ = A_{p}(−p) = 0,

(i) B_{p}(1) = B_{p}(2) = ⋯ = B_{p}(p) = 1, B_{p}(0) = B_{p}(−1) = ⋯ = B_{p}(−p + 2) = 0, B_{p}(−p + 1) = 1 and B_{p}(−p) = 0.

We first derive a relationship between the sequences F_{p} and A_{p}.

Theorem 5. Let n be a positive integer. For positive integers p, we have

F_{p}(n) = A_{p}(n) − A_{p}(n − p).  (2)

Proof. Observe that

F_{p}(n) = \begin{cases}A_{p}(n), & if 0 ≤ n ≤ p; \\ A_{p}(n) − 1, & if p < n ≤ 2p. \end{cases}

So the identity (2) holds for all integers n = 0, 1, 2, …, 2p. Now, assume that for some integer n ≥ 2p, the identity (2) holds for all integers m such that 0 ≤ m ≤ n. By the definition and the inductive hypothesis,

A_{p}(n + 1) − A_{p}(n − p + 1) = A_{p}(n) + A_{p}(n − p + 1) − A_{p}(n − 2p) − A_{p}(n − p + 1)
= A_{p}(n) − A_{p}(n − p) + A_{p}(n − p) − A_{p}(n − 2p)
= F_{p}(n) + F_{p}(n − p)
= F_{p}(n + 1),

showing that the identity (2) holds for n + 1.

Consequently, we can write A_{p}(n) as a sum of the Fibonacci p-numbers.

Corollary 6. Let j be an integer such that 0 ≤ j < p. Then

A_{p}(mp + j) = ∑_{i=0}^{m} F_{p}(ip + j).  (3)

Some special cases of the sequences A_{p} and B_{p} and their first few terms are given in the tables.

• For p = 2, the first 16 terms of A_{2}(n) and B_{2}(n) are

\begin{matrix}n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\ A_{2}(n) & 1 & 1 & 2 & 3 & 5 & 7 & 11 & 16 & 24 & 35 & 52 & 76 & 112 & 164 & 241 & 353 \\ B_{2}(n) & 1 & 1 & 2 & 2 & 4 & 5 & 8 & 11 & 17 & 24 & 36 & 52 & 77 & 112 & 165 & 241 \end{matrix}

• For p = 3, the first 17 terms of A_{3}(n) and B_{3}(n) are

\begin{matrix}n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 \\ A_{3}(n) & 1 & 1 & 1 & 2 & 3 & 4 & 6 & 8 & 11 & 16 & 22 & 30 & 42 & 58 & 80 & 111 & 153 \\ B_{3}(n) & 1 & 1 & 1 & 2 & 2 & 3 & 5 & 6 & 8 & 12 & 16 & 22 & 31 & 42 & 58 & 81 & 111 \end{matrix}

• For p = 4, the first 18 terms of A_{4}(n) and B_{4}(n) are

\begin{matrix}n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 \\ A_{4}(n) & 1 & 1 & 1 & 1 & 2 & 3 & 4 & 5 & 7 & 9 & 12 & 16 & 22 & 29 & 38 & 50 & 67 & 89 \\ B_{4}(n) & 1 & 1 & 1 & 1 & 2 & 2 & 3 & 4 & 6 & 7 & 9 & 12 & 17 & 22 & 29 & 38 & 51 & 67 \end{matrix}

The sequences A_{2}, A_{4}, B_{3} and B_{4} are the sequences \underline{A023435}, \underline{A233522}, \underline{A017819} and \underline{A017830}, respectively.

Some additional properties of the sequences A_{p} and B_{p} are given below. The proof is a straightforward induction and shall be omitted.

Proposition 7. Let n be a nonnegative integer. Then

(i) A_{p}(n) = \begin{cases}A_{p}(n − 1) + A_{p}(n − p − 1) + 1, & if n ≡ 1 (mod p); \\ A_{p}(n − 1) + A_{p}(n − p − 1), & otherwise. \end{cases}

(ii) B_{p}(n) = \begin{cases}B_{p}(n − 1) + B_{p}(n − p − 1) + 1, & if n ≡ 1 (mod p); \\ B_{p}(n − 1) + B_{p}(n − p − 1) − 1, & if n ≡ 2 (mod p); \\ B_{p}(n − 1) + B_{p}(n − p − 1), & otherwise. \end{cases}

Next, we derive a relationship between the sequences A_{p} and B_{p}.

Proposition 8. Let n ≥ 0 be an integer. Then

A_{p}(n) = \begin{cases}B_{p}(n + 1) − 1, if n ≡ 0 (mod p); \\ B_{p}(n + 1),  otherwise. \end{cases}  (4)

Proof. It is easy to see that the base cases of the identity (4) hold. Now assume that, for some integer n ≥ 2p, the identity (4) holds for all integers m such that 0 ≤ m ≤ n. By the definition and the inductive hypothesis, we obtain

A_{p}(n + 1) = A_{p}(n) + A_{p}(n − p + 1) − A_{p}(n − 2p)

= \begin{cases}B_{p}(n + 1) − 1 + B_{p}(n − p + 2) − B_{p}(n − 2p + 1) + 1, & if n ≡ 0 (mod p); \\ B_{p}(n + 1) + B_{p}(n − p + 2) − 1 − B_{p}(n − 2p + 1), & if n ≡ −1 (mod p); \\ B_{p}(n + 1) + B_{p}(n − p + 2) − B_{p}(n − 2p), & otherwise \end{cases}

= \begin{cases}B_{p}(n + 2) − 1, & if n + 1 ≡ 0 (mod p); \\ B_{p}(n + 2), & otherwise, \end{cases}

thereby proving the theorem.

Now we proceed to the main result. For n ≥ p ≥ 2, define the (2p + 1) × (2p + 1) companion matrix Q_{p} = [q_{ij}] where

q_{ij} = \begin{cases}1, & if i = 1, p and j = 1, and i = j + 1; \\ −1, & if i = 2p + 1, j = 1; \\ 0, & otherwise. \end{cases}

For instance, the matrices Q_{p} for p = 2, 3, 4 are as follows:

Q_{2} = [\begin{matrix}1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ −1 & 0 & 0 & 0 & 0 \end{matrix}] , Q_{3} = [\begin{matrix}1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ −1 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}] ,

Q_{4} = [\begin{matrix}1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ −1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}] .

Theorem 9. Let n be a nonnegative integer. Then

Q_{p}^{n} = [\begin{matrix}A_{p}(n + 1) & A_{p}(n) & A_{p}(n − 1) & ⋯ & A_{p}(n − 2p + 1) \\ B_{p}(n − p + 2) & B_{p}(n − p + 3) & B_{p}(n − p + 4) & ⋯ & B_{p}(n − 3p + 2) \\ B_{p}(n − p + 3) & B_{p}(n − p + 4) & B_{p}(n − p + 5) & ⋯ & B_{p}(n − 3p + 3) \\ ⋮ & ⋮ & ... & & ⋮ \\ B_{p}(n) & B_{p}(n − 1) & B_{p}(n − 2) & ⋯ & B_{p}(n − 2p) \\ −A_{p}(n − p) & −A_{p}(n − p − 1) & A_{p}(n − p − 2) & ⋯ & A_{p}(n − 3p) \\ ⋮ & ⋮ & ... & & ⋮ \\ −A_{p}(n − 1) & −A_{p}(n − 2) & A_{p}(n − 3) & ⋯ & A_{p}(n − 2p − 1) \\ −A_{p}(n) & −A_{p}(n − 1) & A_{p}(n − 2) & ⋯ & A_{p}(n − 2p) \end{matrix}] .

Proof. The case n = 1 follows from the initial values of the sequences A_{p} and B_{p}. Write

Q_{p}^{n+1} = Q_{p}^{n}Q_{p}.

By the inductive hypothesis and the definitions of A_{p} and B_{p}, a straightforward matrix multiplication shows that the result holds for n + 1.

For positive integers m, define the (2p + 1) × (2p + 1) matrix R_{p,m} by

R_{p,m} = [\begin{matrix}H_{p}(m) & H_{p}(m − 1) & H_{p}(m − 2) & ⋯ & H_{p}(m − 2p) \\ H_{p}(m − 1) & H_{p}(m − 2) & H_{p}(m − 3) & ⋯ & H_{p}(m − 2p − 1) \\ H_{p}(m − 2) & H_{p}(m − 3) & H_{p}(m − 4) & ⋯ & H_{p}(m − 2p − 2) \\ ⋮ & ⋮ & ... & & ⋮ \\ H_{p}(m − 2p) & H_{p}(m − 2p − 1) & H_{p}(m − 2p − 2) & ⋯ & H_{p}(m − 4p) \end{matrix}]

where H_{p} is an element of H(p).

Theorem 10. Let n ≥ 0 be an integer. For every positive integer m, we have

R_{p,m}Q_{p}^{n} = R_{p,m+n}.  (5)

Proof. The case n = 0 is trivial. Now assume that the identity (5) is true for an integer n ≥ 0. Since

R_{p,m}Q_{p}^{n+1} = (R_{p,m}Q_{p}^{n}) Q_{p},

by the inductive hypothesis and the definition of H_{p}, a straightforward matrix multiplication shows that the result holds for n + 1.

Equating the (1, 1)-entry on both sides of the equation (5), we obtain the following corollary.

Corollary 11. Let m, n > 0 be two integers. For a sequence H_{p} ∈ H(p), we have

H_{p}(m+n) = H_{p}(m)A_{p}(n+1)+∑_{i=1}^{p−1} H_{p}(m−i)B_{p}(n−p+i+1)−∑_{i=0}^{p} H_{p}(m−p−i)A_{p}(n−p+i).

Note that, since L_{p,k}, A_{p}, B_{p} ∈ H(p), the identity of Corollary 11 holds for each of these sequences as well.

Lastly, we derive the analogous version of Honsberger’s formula for the generalized Leonardo p-numbers.

Lemma 12. Let n be a positive integer. Then

(k + 1) ∑_{i=0}^{p} F_{p}(n + i) = L_{p,k}(n + 2p − 1) + k.

Proof. Using the sum of Fibonacci p-numbers formula (see, e.g., [7])

∑_{i=1}^{n} F_{p}(i) = F_{p}(n + p + 1) − 1,

we get that

∑_{i=0}^{p} F_{p}(n + i) = ∑_{i=1}^{n+p} F_{p}(i) − ∑_{i=1}^{n−1} F_{p}(i)
= F_{p}(n + 2p + 1) − F_{p}(n + p)
= F_{p}(n + 2p).

Using Theorem 2, we obtain

(k + 1) ∑_{i=0}^{p} F_{p}(n + i) = (k + 1)F_{p}(n + 2p) = L_{p,k}(n + 2p − 1) + k,

as desired.

Theorem 13. Let m and n be two positive integers. Then

L_{p,k}(m)L_{p,k}(n + 1) + ∑_{i=1}^{p} L_{p,k}(m − j)L_{p,k}(n − p + i)
= (k + 1)L_{p,k}(m + n + 1) − k(L_{p,k}(m + p) + L_{p,k}(n + p + 1)) + pk^{2} + k.

Proof. By Theorem 2, write

L_{p,k}(m)L_{p,k}(n) = (k + 1)^{2}F_{p}(m + 1)F_{p}(n + 1) − k(k + 1)(F_{p}(m + 1) + F_{p}(n + 1)) + k^{2}. Applying Lemma 12 and Honsberger’s formula for the Fibonacci p-numbers [9], we have

L_{p,k}(m)L_{p,k}(n + 1) + ∑_{i=1}^{p} L_{p,k}(m − j)L_{p,k}(n − p + i)

= (k + 1)^{2} (F_{p}(m + 1)F_{p}(n + 2) + ∑_{i=1}^{p} F_{p}(m − i + 1)F_{p}(n − p + i + 1))

− k(k + 1) (F_{p}(m + 1) + F_{p}(n + 2) + ∑_{i=1}^{p} F_{p}(m − i + 1)F_{p}(n − p + i + 1))

+ (p + 1)k^{2}

= (k + 1)^{2}F_{p}(m + n + 2) − k(k + 1) (∑_{i=0}^{p} F_{p}(m − i + 1) + ∑_{i=1}^{p+1} F_{p}(n − p + i + 1))

+ (p + 1)k^{2}

= (k + 1)(L_{p,k}(m + n + 1) + k) − k (L_{p,k}(m + p) + L_{p,k}(n + p + 1) + 2k)

+ (p + 1)k^{2}

= (k + 1)L_{p,k}(m + n + 1) − k(L_{p,k}(m + p) + L_{p,k}(n + p + 1)) + pk^{2} + k, as desired.

Substuting k = p in Theorem 13 and using Lemma 12, we obtain Theorem 1 of [8].

4 Acknowledgment

The authors are truly grateful to the anonymous referee for his/her helpful comments and suggestions.

References

[1] Y. Alp and E. G. Kocer, Some properties of Leonardo numbers, Konuralp J. Math. 9 (2021), 183–189.

[2] P. Catarino and A. Borges, On Leonardo numbers, Acta Math. Univ. Comenian. (N.S.) 89 (2019), 75–86.

[3] P. Catarino and A. Borges, A note on incomplete Leonardo numbers, Integers 20 (2020), #A43.

[4] K. Kuhapatanakul and J. Chobsorn, On the generalized Leonardo numbers, Integers 22 (2022), #A48.

[5] A. G. Shannon, A note on generalized Leonardo numbers, Notes Number Theory Discrete Math. 25 (2019), 97–101.

[6] N. J. A. Sloane et al., The On-Line Encyclopedia of Integer Sequences, 2024. Available

   at https://oeis.org.

[7] A. P. Stakhov, Fibonacci matrices, a generalization of the Cassini formula, and a new coding theory, Chaos Solitons Fractals 30 (2006), 56–66.

[8] N. Tuglu, E. G. Kocer, and A. Stakhov, Bivariate Fibonacci like p-polynomials, Appl. Math. Comput. 217 (2011), 10239–10249.

[9] E. Tan and H. H. Leung, On Leonardo p-numbers, Integers 23 (2023), #A7.

2020 Mathematics Subject Classification: Primary 11B37; Secondry 11B39. Keywords: Leonardo number, Fibonacci number, Leonardo p-number.

(Concerned with sequences \underline{A000045}, \underline{A000930}, \underline{A001595}, \underline{A017819}, \underline{A017830}, \underline{A023435}, \underline{A111314}, \underline{A233522}, and \underline{A362255}.)

Received October 2 2023; revised versions received October 9 2023; March 18 2024; March 21 2024. Published in Journal of Integer Sequences, April 11 2024.

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