On (k1A1, k2A2, k3A3)-Edge Colourings in Graphs and Generalized Jacobsthal Numbers

Piejko, Krzysztof; Trojnar-Spelina, Lucyna

doi:10.2478/amsil-2025-0006

Full Article

1.

Introduction and preliminary results

We use the standard definitions and notations of the graph theory, see [4] and [6]. Let us recall definitions of several very important sequences in the numbers theory. The Fibonacci sequence (F_n) is defined by F₀ = 0, F₁ = 1 and F_n = F_n₋₁ + F_n₋₂, for n ≥ 2. The Pell sequence (P_n) is also defined by recurrence relation P_n = 2P_n₋₁ + P_n₋₂, for n ≥ 2 with the initial conditions P₀ = 0, P₁ = 1. The Jacobsthal sequence (J_n) is defined recursively as follows J_n = J_n₋₁ + 2J_n₋₂, for n ≥ 2 with the initial conditions J₀ = 0, J₁ = 1. The first few Fibonacci, Pell and Jacobsthal numbers are 0, 1, 1, 2, 3, 5, 8, . . . , 0, 1, 2, 5, 12, 29, 70, . . . and 0, 1, 1, 3, 5, 11, 21, . . . , respectively.

In recent years many interesting generalizations of above mentioned classical sequences appeared in the mathematical literature. For example generalizations of the Pell numbers were introduced and studied in [8],[9],[10], and in [13], [15],[16],[17],[18], [20]. Moreover in [16] generalized Jacobsthal numbers were considered.

In this paper we study a new kind of generalization of Jacobsthal numbers in the distance sense, i.e., generalization by the k-th order linear recurrence relation. This concept of generalization of Jacobsthal numbers is inspired by results given in [1],[2],[3], [5], [15], [17] and [19] where generalizations of Fibonacci numbers, Lucas numbers and Pell numbers were introduced.

For fixed integers k ≥ 1 and n ≥ 0 by J⁽ⁱ⁾(k, n) we denote the (2, k)-distance Jacobsthal numbers of the i-th kind, i = 1, 2, i.e., numbers defined as follows (1) $J^{(i)} (k, n) = J^{(i)} (k, n - k) + 2 J^{(i)} (k, n - 2) for n \geq k + 1$ {J^{\left(i \right)}}\left({k,n} \right) = {J^{\left(i \right)}}\left({k,n - k} \right) + 2{J^{\left(i \right)}}\left({k,n - 2} \right)\,\,\,\,\,{\rm{for}}\,\,\,\,\,n \ge k + 1 with initial conditions J⁽ⁱ⁾(k, 0) = 0, J⁽ⁱ⁾(k, 1) = 1 and for 2 ≤ n ≤ k: $J^{(1)} (k, n) = \begin{array}{l} \begin{array}{l} 0 & if n is even, \\ 2^{\frac{n - 1}{2}} & if n is odd, \end{array} & J^{(2)} (k, n) = 2^{\frac{n - 1}{2}⌋} . \end{array}$ {J^{\left(1 \right)}}\left({k,n} \right) = \left\{{\matrix{{\matrix{0 \hfill & {{\rm{if}}\,n\,{\rm{is}}\,{\rm{even}},} \hfill \cr {{2^{{{n - 1} \over 2}}}} \hfill & {{\rm{if}}\,n\,{\rm{is}}\,{\rm{odd}},} \hfill \cr}} \hfill & {{J^{(2)}}(k,n) = {2^{\left\lfloor {{{n - 1} \over 2}} \right\rfloor}}.} \hfill \cr}} \right.

In the following tables, we present several initial terms of the (2, k)-distance Jacobsthal sequences J⁽ⁱ⁾(k, n), i = 1, 2, for special values of k and n.

Table 1.

The (2, k)-distance Jacobsthal numbers of the first kind J⁽¹⁾(k, n)

n	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
J⁽¹⁾(1, n)	1	1	3	5	11	21	43	85	171	341	683	1365	2731	5461	10923
J⁽¹⁾(2, n)	1	0	3	0	9	0	27	0	81	0	243	0	729	0	2187
J⁽¹⁾(3, n)	1	0	2	1	4	4	9	12	22	33	56	88	145	232	378
J⁽¹⁾(4, n)	1	0	2	0	5	0	12	0	29	0	70	0	169	0	408
J⁽¹⁾(5, n)	1	0	2	0	4	1	8	4	16	12	33	32	70	80	152
J⁽¹⁾(6, n)	1	0	2	0	4	0	9	0	20	0	44	0	97	0	214
J⁽¹⁾(7, n)	1	0	2	0	4	0	8	1	16	4	32	12	64	32	129

Table 2.

The (2, k)-distance Jacobsthal numbers of the second kind J⁽²⁾(k, n)

n	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
J⁽²⁾(1, n)	1	1	3	5	11	21	43	85	171	341	683	1365	2731	5461	10923
J⁽²⁾(2, n)	1	1	3	3	9	9	27	27	81	81	243	243	729	729	2187
J⁽²⁾(3, n)	1	1	2	3	5	8	13	21	34	55	89	144	233	377	610
J⁽²⁾(4, n)	1	1	2	2	5	5	12	12	29	29	70	70	169	169	408
J⁽²⁾(5, n)	1	1	2	2	4	5	9	12	20	28	45	65	102	150	232
J⁽²⁾(6, n)	1	1	2	2	4	4	9	9	20	20	44	44	97	97	214
J⁽²⁾(7, n)	1	1	2	2	4	4	8	9	17	20	36	44	76	96	161

From the definition of the (2, k)-distance Jacobsthal numbers it follows immediately that J⁽¹⁾(1, n) = J⁽²⁾(1, n) = J_n. The numbers J⁽ⁱ⁾(2, n), i = 1, 2, are the elements of geometrical sequence, namely $J^{(1)} (2, n) = 3^{\frac{n - 1}{2}}$ {J^{\left(1 \right)}}(2,n) = {3^{{{n - 1} \over 2}}} for odd n and $J^{(2)} (2, n) = 3^{\frac{n - 1}{2}⌋}$ {J^{\left(2 \right)}}(2,n) = {3^{\left\lfloor {{{n - 1} \over 2}} \right\rfloor}} . Moreover if k is even and n is odd then J⁽²⁾(k, n) = J⁽²⁾(k, n + 1).

For k = 3 we have the following relationships between J⁽¹⁾(k, n), J⁽²⁾(k, n) and the Fibonacci numbers F_n while for k = 4 we have their connections with the Pell numbers P_n.

Theorem 1

Let n be an integer. Then

(i)
J⁽¹⁾(3, n) = F_n₋₁ − (−1)ⁿ for n ≥ 1,
(ii)
J⁽²⁾(3, n) = F_n for n ≥ 0,
(iii)
J⁽¹⁾(4, 2n − 1) = P_n for n ≥ 1,
(iv)
J⁽²⁾(4, 2n − 1) = J⁽²⁾(4, 2n) = P_n for n ≥ 1.

Proof

In the proof of formula (i) we use the induction on n. For n = 1 or n = 2 the result follows immediately by the definitions of J⁽¹⁾(k, n) and the Fibonacci numbers F_n.

Assume now that formula (i) is true for t = 1, 2, . . . , n with arbitrary n ≥ 2. We will prove that J⁽¹⁾(3, n + 1) = F_n − (−1)ⁿ⁺¹. By the induction hypothesis and the definitions of J⁽¹⁾(k, n) and F_n, we have $\begin{array}{l} J^{(1)} (3, n + 1) & = J^{(1)} (3, n - 2) + 2 J^{(1)} (3, n - 1) \\ = F_{n - 3} - {(- 1)}^{n - 2} + 2 (F_{n - 2} - {(- 1)}^{n - 1}) \\ = 2 F_{n - 2} + F_{n - 3} + {(- 1)}^{n} = F_{n - 2} + F_{n - 1} + (- 1^{n}) \\ = F_{n} - {(- 1)}^{n + 1}, \end{array}$ \matrix{{{J^{\left(1 \right)}}\left({3,n + 1} \right)} \hfill & {= {J^{\left(1 \right)}}\left({3,n - 2} \right) + 2{J^{\left(1 \right)}}\left({3,n - 1} \right)} \hfill \cr {} \hfill & {= {F_{n - 3}} - {{(- 1)}^{n - 2}} + 2({F_{n - 2}} - {{(- 1)}^{n - 1}})} \hfill \cr {} \hfill & {= 2{F_{n - 2}} + {F_{n - 3}} + {{(- 1)}^n} = {F_{n - 2}} + {F_{n - 1}} + (- {1^n})} \hfill \cr {} \hfill & {= {F_n} - {{(- 1)}^{n + 1}},} \hfill \cr} which ends the proof of (i). To prove (ii)–(iv) we can use the same method.

The following theorem shows the relation between numbers J⁽ⁱ⁾(k, n) for i = 1, 2.

Theorem 2

Let k ≥ 2, n ≥ 1 be integers. Then

(i)
J⁽²⁾(k, n) = J⁽¹⁾(k, n) + J⁽¹⁾(k, n − 1),
(ii)
$J^{(1)} (k, n) = \sum_{i = 0}^{n} {(- 1)}^{i} J^{(2)} (k, n - i)$ {J^{(1)}}\left({k,n} \right) = \sum\limits_{i = 0}^n {{{(- 1)}^i}{J^{\left(2 \right)}}(k,n - i)} .

Proof

In the proof of formula (i) we use the induction on n. Let n = 1, 2, . . . , k − 1. For k = 2 the result is obvious. If k ≥ 3 then we have (i) from initial conditions for J⁽¹⁾(k, n) and J⁽²⁾(k, n). Assume now that n ≥ k − 1 and that the equality (i) holds for all integers k ≤ t ≤ n. We shall prove that it is true for the integer t = n + 1. Using (1) and the induction hypothesis, we obtain that $\begin{array}{l} J^{(2)} (k, n + 1) = J^{(2)} (k, n + 1 - k) + 2 J^{(2)} (k, n - 1) \\ = J^{(1)} (k, n + 1 - k) + J^{(1)} (k, n - k) + 2 (J^{(1)} (k, n - 1) + J^{(1)} (k, n - 2)) \\ = J^{(1)} (k, n + 1 - k) + 2 J^{(1)} (k, n - 1) + J^{(1)} (k, n - k) + 2 J^{(1)} (k, n - 2) \\ = J^{(1)} (k, n + 1) + J^{(1)} (k, n), \end{array}$ \matrix{{{J^{\left(2 \right)}}\left({k,n + 1} \right) = {J^{\left(2 \right)}}\left({k,n + 1 - k} \right) + 2{J^{\left(2 \right)}}\left({k,n - 1} \right)} \hfill \cr {\,\,\,\,\,\,\,\, = {J^{\left(1 \right)}}\left({k,n + 1 - k} \right) + {J^{\left(1 \right)}}\left({k,n - k} \right) + 2\left({{J^{\left(1 \right)}}\left({k,n - 1} \right) + {J^{\left(1 \right)}}\left({k,n - 2} \right)} \right)} \hfill \cr {\,\,\,\,\,\,\,\, = {J^{\left(1 \right)}}\left({k,n + 1 - k} \right) + 2{J^{\left(1 \right)}}\left({k,n - 1} \right) + {J^{\left(1 \right)}}\left({k,n - k} \right) + 2{J^{\left(1 \right)}}\left({k,n - 2} \right)} \hfill \cr {\,\,\,\,\,\,\,\, = {J^{\left(1 \right)}}\left({k,n + 1} \right) + {J^{\left(1 \right)}}\left({k,n} \right),} \hfill \cr} which ends the proof of (i). The equality (ii) can be proved in the same manner.

In this paper we give the graph interpretations of considered two types of the (2, k)-distance Jacobsthal numbers with respect to a special edge colouring of a graph. Next, using these interpretations, we obtain several identities and direct formulas for them. We also give matrix representations for J⁽ⁱ⁾(k, n), i = 1, 2.

2.

Graph interpretations

Let us consider an edge coloured graph G where the set of colours is {A₁, A₂, A₃}. For α ∈ {A₁, A₂, A₃} a subgraph of G will be said α-monochromatic if all its edges are coloured by colour α. We will write α(xy) if the edge xy of the graph has the colour α.

In [17], Trojnar-Spelina and Włoch defined so called (k₁A₁, k₂A₂, k₃A₃)-edge colouring by monochromatic paths in a graph G where k_i ≥ 1, i = 1, 2, 3, are integers, as a generalization of the edge-colouring introduced and studied by Piejko and Włoch in [15]. Let us recall that the (k₁A₁, k₂A₂, k₃A₃)-edge colouring by monochromatic paths in a graph G is defined in such a way that every maximal, with respect to the set inclusion, A_i-monochromatic subgraph of G, can be partitioned into edge-disjoint paths of the length k_i, i = 1, 2, 3 (see [17]). Many interesting properties of (k₁A₁, k₂A₂, k₃A₃)-edge colouring by monochromatic paths were given in [17]. For example, it was proved that the number of (kA₁, kA₂, 2A₃)-edge colourings by monochromatic paths is closely related to (2, k)-distance Pell numbers of the i-th kind, i = 1, 2, defined recursively as follows $P^{(i)} (k, n) = 2 P^{(i)} (k, n - k) + P^{(i)} (k, n - 2) for n \geq k$ {P^{\left(i \right)}}\left({k,n} \right) = 2{P^{\left(i \right)}}\left({k,n - k} \right) + {P^{\left(i \right)}}\left({k,n - 2} \right)\,\,\,\,\,{\rm{for}}\,\,\,\,\,n \ge k with initial conditions P⁽ⁱ⁾(k, 0) = 0, P⁽ⁱ⁾(k, 1) = 1 and for 2 ≤ n ≤ k − 1: $P^{(1)} (k, n) = \begin{array}{l} \begin{array}{l} 0 & if n is even, \\ 1 & if n is odd, \end{array} & and & P^{(2)} (k, n) = 1 . \end{array}$ {P^{\left(1 \right)}}\left({k,n} \right) = \left\{{\matrix{{\matrix{0 \hfill & {{\rm{if}}\,n\,{\rm{is}}\,{\rm{even}},} \hfill \cr 1 \hfill & {{\rm{if}}\,n\,{\rm{is}}\,{\rm{odd}},} \hfill \cr}} \hfill & {{\rm{and}}} \hfill & {{P^{(2)}}(k,n) = 1.} \hfill \cr}} \right.

In this section, we give graph interpretations for the (2, k)-distance Jacobsthal numbers J⁽ⁱ⁾(k, n) in terms of (k₁A₁, k₂A₂, k₃A₃)-edge colouring by monochromatic paths with k₁ = k, k ≥ 1 and k₂ = k₃ = 2. We will use standard notation 𝒫_n for a path with n vertices.

Theorem 3

Let k ≥ 1, n ≥ 2 be integers. The number of all (kA₁, 2A₂, 2A₃)-edge colourings by monochromatic paths of the graph 𝒫_n is equal to J⁽¹⁾(k, n).

Proof

Let k ≥ 1, n ≥ 2 be integers and let V (𝒫_n) = {x₁, x₂, . . . , x_n} with the numbering of vertices in the natural fashion. Then E(𝒫_n) = {x₁x₂, x₂x₃, . . . , x_n₋₁x_n}. The number of all (kA₁, 2A₂, 2A₃)-edge colourings by monochromatic paths of the graph 𝒫_n will be denoted by σ(k, n). One can easy verify that σ(k, n) = J⁽¹⁾(k, n) for n = 2, . . . , k + 2.

Assume now that n ≥ k + 3. Denote by σ_A₁ (k, n), σ_A₂ (k, n), σ_A₃ (k, n), the number of (kA₁, 2A₂, 2A₃)-edge colourings by monochromatic paths of the graph 𝒫_n in which the colour of the last edge is determined respectively as follows: A₁(x_n₋₁x_n), A₂(x_n₋₁x_n), A₃(x_n₋₁x_n). It is clear that σ_A₁ (k, n) is equal to the number of all (kA₁, 2A₂, 2A₃)-edge colourings by monochromatic paths of the graph 𝒫_n−k and σ_A₂ (k, n), σ_A₃ (k, n) are equal to the number of all (kA₁, 2A₂, 2A₃)-edge colourings by monochromatic paths of the graph 𝒫_n−2. In other words σ_A₁ (k, n) = σ(k, n − k), σ_A₂ (k, n) = σ(k, n − 2) and σ_A₃ (k, n) = σ(k, n − 2). From the above we have $σ (k, n) = σ_{A_{1}} (k, n) + σ_{A_{2}} (k, n) + σ_{A_{3}} (k, n) = σ (k, n - k) + 2 σ (k, n - 2) .$ \sigma \left({k,n} \right) = {\sigma_{{A_1}}}\left({k,n} \right) + {\sigma_{{A_2}}}\left({k,n} \right) + {\sigma_{{A_3}}}\left({k,n} \right) = \sigma \left({k,n - k} \right) + 2\sigma \left({k,n - 2} \right). Taking into account the initial conditions we obtain that σ(k, n) = J⁽¹⁾(k, n) for all n ≥ 2. The proof of Theorem 3 is completed.

Hence we obtain the following graph interpretation of Jacobsthal numbers.

Corollary 1

Let n ≥ 2 be an integer. The number of all (A₁, 2A₂, 2A₃)-edge colourings of the graph 𝒫_n is equal to J_n.

Before we give the graph interpretation for the (2, k)-distance Jacobsthal numbers of the second kind J⁽²⁾(k, n) we introduce a quasi (k₁A₁, k₂A₂, k₃A₃)-edge colouring by monochromatic paths in a graph 𝒫_n. Let E(𝒫_n) = {e₁, e₂, . . . , e_n₋₁} be numbered in the natural fashion. We say that the graph 𝒫_n is a quasi (k₁A₁, k₂A₂, k₃A₃)-edge coloured by monochromatic paths if the last r edges of 𝒫_n are uncoloured where 0 ≤ r ≤ t − 1, t = min{k₁, k₂, k₃} and the subpath 𝒫^∗ ⊂ 𝒫_n induced by {e₁, e₂, . . . , e_n_−r−1} is (k₁A₁, k₂A₂, k₃A₃)-edge coloured by monochromatic paths.

From above immediately follows that (k₁A₁, k₂A₂, k₃A₃)-edge coloured 𝒫_n is also quasi (k₁A₁, k₂A₂, k₃A₃)-edge coloured.

Using the same manner as that in the proof of Theorem 3 we can prove that the number of all quasi (kA₁, 2A₂, 2A₃)-edge colourings by monochromatic paths of the graph 𝒫_n closely corresponds to the (2, k)-distance Jacobsthal numbers of the second kind J⁽²⁾(k, n). Namely we have

Theorem 4

Let k ≥ 1, n ≥ 2 be integers. The number of all quasi (kA₁, 2A₂, 2A₃)-edge colourings by monochromatic paths of the graph 𝒫_n is equal to J⁽²⁾(k, n).

These graph interpretations will be used to derive the direct formulas for the (2, k)-distance Jacobsthal numbers J⁽ⁱ⁾(k, n), i = 1, 2.

Before that, we need to deduce certain preliminary results. Let k ≥ 1, n ≥ 2 and $0 \leq t \leq \frac{n - 1}{2}⌋$ 0 \le t \le \left\lfloor {{{n - 1} \over 2}} \right\rfloor be integers. Let p_k(n, t) denote the number of (kA₁, 2A₂, 2A₃)-edge colourings by monochromatic paths of the graph 𝒫_n such that there are exactly t monochromatic paths of the length 2 coloured by A₂ or A₃. This means that in such edge colouring 2t₁ edges have colour A₂, 2t₂ edges have colour A₃ where t₁ + t₂ = t and other n − 1 − 2t edges have colour A₁.

Theorem 5

Let n ≥ 2 and $0 \leq t \leq \frac{n - 1}{2}⌋$ 0 \le t \le \left\lfloor {{{n - 1} \over 2}} \right\rfloor be integers. Then $p_{1} (n, t) = (\begin{matrix} n - 1 - t \\ t \end{matrix}) 2^{t} .$ {p_1}\left({n,t} \right) = \left({\matrix{{n - 1 - t} \cr t \cr}} \right){2^t}.

Proof

Let us consider the graph 𝒫_n with the (A₁, 2A₂, 2A₃)-edge colouring by monochromatic paths in which there are exactly t monochromatic paths of the length 2 coloured by the colour A₂ or A₃. This edge colouring is related to a certain partition of the set E(𝒫_n) into t edge-disjoint A_i monochromatic paths of the length 2 for i = 2, 3 and n − 1 − 2t monochromatic paths of the length 1 for i = 1. We denote a number of all such partitions by δ(n, t). It is easy to observe that δ(n, t) equals to the number of all permutations with repetitions of n − 1 − t elements that are equal to 1 or 2 where the number 1 is repeated n − 1 − t times and the number 2 is repeated t times. Therefore, $δ (n, t) = P_{n - 1 - t}^{t, n - 1 - 2 t}$ \delta \left({n,t} \right) = P_{n - 1 - t}^{t,n - 1 - 2t} , where the symbol $P_{n}^{n_{1}, n_{2}}$ P_n^{{n_1},{n_2}} denotes the number of all permutations with repetitions of the length n of two elements, where first of these elements is repeated n₁ times, the second is repeated n₂ times and n₁ + n₂ = n. Consequently we have $δ (n, t) = \frac{(n - 1 - t)!}{t! (n - 1 - 2 t)!} = (\begin{matrix} n - 1 - t \\ t \end{matrix}) .$ \delta \left({n,t} \right) = {{\left({n - 1 - t} \right)!} \over {t!\left({n - 1 - 2t} \right)!}} = \left({\matrix{{n - 1 - t} \cr t \cr}} \right). Every of this partitions generates 2^t (A₁, 2A₂, 2A₃)-edge colourings by monochromatic paths. Therefore, the desired result follows.

Theorem 6

Let k ≥ 1, n ≥ 2 and $0 \leq t \leq \frac{n - 1}{2}⌋$ 0 \le t \le \left\lfloor {{{n - 1} \over 2}} \right\rfloor be integers and let $\frac{n - 1 - 2 t}{k} \in N \cup {0}$ {{n - 1 - 2t} \over k} \in N \cup \{0\} . Then for s = 0, 1, . . . , n − 1 we have $p_{k} (n, t) = p_{k - s} (n - \frac{s (n - 1 - 2 t)}{k}, t) .$ {p_k}(n,t) = {p_{k - s}}\left({n - {{s\left({n - 1 - 2t} \right)} \over k},t} \right).

Proof

Consider the (kA₁, 2A₂, 2A₃)-edge colouring by monochromatic paths of the graph 𝒫_n in which there are exactly t monochromatic paths of the length 2 coloured by the colour A₂ or A₃. Consequently, there are exactly $\frac{n - 1 - 2 t}{k}$ {{n - 1 - 2t} \over k} monochromatic paths of the length k coloured by A₁ in this edge colouring. If every of these paths is shorted to the A₁-monochromatic paths of the length k − s, then we obtain a ((k − s)A₁, 2A₂, 2A₃)-edge colouring by monochromatic paths of the graph $P_{n - s \frac{n - 1 - 2 t}{k}}$ {{\mathcal P}_{n - s{{n - 1 - 2t} \over k}}} , in which there are $\frac{n - 1 - 2 t}{k}$ {{n - 1 - 2t} \over k} monochromatic paths of the length k − s coloured by A₁ and the number of monochromatic paths of the length 2 coloured by A_i, i = 2, 3, remains the same as in the starting edge colouring. Therefore the proof is completed.

Putting s = k − 1 in Theorem 6 we have $p_{k} (n, t) = p_{1} (n - \frac{(k - 1) (n - 1 - 2 t)}{k}, t)$ {p_k}\left({n,t} \right) = {p_1}\left({n - {{\left({k - 1} \right)\left({n - 1 - 2t} \right)} \over k},t} \right) and so using Theorem 5 we obtain the following

Corollary 2

Let k ≥ 1, n ≥ 2 and $0 \leq t \leq \frac{n - 1}{2}⌋$ 0 \le t \le \left\lfloor {{{n - 1} \over 2}} \right\rfloor be integers and let $\frac{n - 1 - 2 t}{k} \in N \cup {0}$ {{n - 1 - 2t} \over k} \in N \cup \{0\} . Then $p_{k} (n, t) = (\begin{matrix} \frac{1}{k} [n - 1 + t (k - 2)] \\ t \end{matrix}) 2^{t} .$ {p_k}(n,t) = \left({\matrix{{{1 \over k}[n - 1 + t(k - 2)]} \cr t \cr}} \right){2^{t}}.

Let us consider again such (kA₁, 2A₂, 2A₃)-edge colouring by monochromatic paths of the graph 𝒫_n, that there are exactly t monochromatic paths of the length 2 coloured by A₂ or A₃, i.e., in such edge colouring 2t₁ edges have colour A₂, 2t₂ edges have colour A₃ where t₁ +t₂ = t and other n−1−2t edges have colour A₁. We can observe that such (kA₁, 2A₂, 2A₃)-edge colouring by monochromatic paths of the graph 𝒫_n exists if and only if a number $\frac{n - 1 - 2 t}{k}$ {{n - 1 - 2t} \over k} is nonnegative integer. For given k ≥ 1 and n ≥ 2 we introduce the following notation: $I_{k}^{n} = {t \in N \cup {0} : \frac{n - 1 - 2 t}{k} \in N \cup {0}} .$ I_k^n = \{t \in N \cup \{0\} :{{n - 1 - 2t} \over k} \in N \cup \{0\} \} .

Note that for example: $I_{10}^{3} = 0, 3\}$ I_{10}^3 = \left\{{0,3} \right\} , $I_{14}^{4} = \emptyset$ I_{14}^4 = \emptyset and $I_{1}^{n} = 0, 1, 2, \dots, \frac{n - 1}{2}⌋\}$ I_1^n = \left\{{0,1,2,\; \ldots ,\;\left\lfloor {{{n - 1} \over 2}} \right\rfloor} \right\} for all positive integers n. Moreover, if $I_{k}^{n} \neq \emptyset$ I_k^n \ne \emptyset then p_k(n, t) ≠ 0 for all $t \in I_{k}^{n}$ t \in I_k^n and otherwise p_k(n, t) = 0 for all t ≥ 0.

Now we are able to present the following direct formula for the numbers J⁽¹⁾(k, n).

Theorem 7

Let k ≥ 1, n ≥ 1 and $0 \leq t \leq \frac{n - 1}{2}⌋$ 0 \le t \le \left\lfloor {{{n - 1} \over 2}} \right\rfloor be integers. Then $J^{(1)} (k, n) = \sum_{t \in I_{k}^{n}} (\begin{matrix} \frac{1}{k} [n - 1 + t (k - 2)] \\ t \end{matrix}) 2^{t} if I_{k}^{n} \neq \emptyset$ {J^{\left(1 \right)}}(k,n) = \sum\limits_{t \in I_k^n} {\left({\matrix{{{1 \over k}[n - 1 + t(k - 2)]} \cr t \cr}} \right){2^t}\,\,\,\,\,if\,\,\,\,\,I_k^n \ne \emptyset} and $J^{(1)} (k, n) = 0 if I_{k}^{n} = \emptyset .$ {J^{\left(1 \right)}}\left({k,n} \right) = 0\,\,\,\,\,if\,\,\,\,\,I_k^n = \emptyset .

Proof

Using Theorem 3 we have that J⁽¹⁾(k, n) is equal to the number of all (kA₁, 2A₂, 2A₃)-edge colourings by monochromatic paths of the graph 𝒫_n. Therefore for given k ≥ 1, n ≥ 1 it can be expressed as follows $J^{(1)} (k, n) = \sum_{t \in I_{k}^{n}} p_{k} (n, t) .$ {J^{\left(1 \right)}}(k,n) = \sum\limits_{t \in I_k^n} {{p_k}\left({n,t} \right).} Let us recall that if $I_{k}^{n} \neq \emptyset$ I_k^n \ne \emptyset then p_k(n, t) = 0 for all t ≥ 0 and therefore in this case the result follows. Otherwise, we have p_k(n, t) ≠ 0 for all $t \in I_{k}^{n}$ t \in I_k^n , so an application of Corollary 2 enables us to deduce the desired equality.

Putting k = 1 in Theorem 7, we obtain the following well-known direct formula for Jacobsthal numbers.

Corollary 3

Let n ≥ 1 and $0 \leq t \leq \frac{n - 1}{2}⌋$ 0 \le t \le \left\lfloor {{{n - 1} \over 2}} \right\rfloor be integers. Then $J_{n} = \sum_{t = 1}^{\frac{n - 1}{2}⌋} (\begin{matrix} n - 1 - t \\ t \end{matrix}) 2^{t} .$ {J_n} = \sum\limits_{t = 1}^{\left\lfloor {{{n - 1} \over 2}} \right\rfloor} {\left({\matrix{{n - 1 - t} \cr t \cr}} \right){2^t}.}

To derive the direct formula for J⁽²⁾(k, n) we need new notations. Let k ≥ 1, n ≥ 2 and $0 \leq t \leq \frac{n - 1}{2}⌋$ 0 \le t \le \left\lfloor {{{n - 1} \over 2}} \right\rfloor be integers. By q_k(n, t) we denote the number of quasi (kA₁, 2A₂, 2A₃)-edge colourings by monochromatic paths of the graph 𝒫_n such that exactly t monochromatic paths are coloured by A₂ or A₃. Let us note that in this edge colouring the number of A₁-monochromatic paths is equal to $\frac{n - 1 - 2 t}{k}⌋$ \left\lfloor {{{n - 1 - 2t} \over k}} \right\rfloor and consequently q₁(n, t) = p₁(n, t) so by Theorem 5 we have (2) $q_{1} (n, t) = (\begin{matrix} n - 1 - t \\ t \end{matrix}) 2^{t} .$ {q_1}\left({n,t} \right) = \left({\matrix{{n - 1 - t} \cr t \cr}} \right){2^t}.

Theorem 8

Let k ≥ 1, n ≥ 2 and $0 \leq t \leq \frac{n - 1}{2}⌋$ 0 \le t \le \left\lfloor {{{n - 1} \over 2}} \right\rfloor be integers and s ∈ N. Then $q_{k} (n, t) = q_{s} (1 + 2 t + \frac{n - 1 - 2 t}{k}⌋ s, t) .$ {q_k}\left({n,t} \right) = {q_s}\left({1 + 2t + \left\lfloor {{{n - 1 - 2t} \over k}} \right\rfloor s,t} \right).

Proof

Let us consider the quasi (kA₁, 2A₂, 2A₃)-edge colouring by monochromatic paths of the graph 𝒫_n in which exactly t monochromatic paths of the length 2 have colour A₂ or A₃. Therefore there are $\frac{n - 1 - 2 t}{k}⌋$ \left\lfloor {{{n - 1 - 2t} \over k}} \right\rfloor A₁-monochromatic paths of the length k in this edge-colouring. If, for a given positive integer n, every of these paths is replaced by A₁-monochromatic paths of the length s, then we obtain a quasi (sA₁, 2A₂, 2A₃)-edge colouring by monochromatic paths of the graph $P_{\frac{n - 1 - 2 t}{k}⌋ s}$ {{\mathcal P}_{\left\lfloor {{{n - 1 - 2t} \over k}} \right\rfloor s}} in which the number of monochromatic paths of the length 2 coloured by A_i, i = 2, 3 remains the same as in the starting edge colouring. The proof is completed.

Putting s = 1 in Theorem 8 and using the relation (2) we obtain the following result

Corollary 4

Let k ≥ 1, n ≥ 2 and $0 \leq t \leq \frac{n - 1}{2}⌋$ 0 \le t \le \left\lfloor {{{n - 1} \over 2}} \right\rfloor be integers. Then $q_{k} (n, t) = (\begin{matrix} t + \frac{n - 1 - 2 t}{t}⌋ \\ t \end{matrix}) 2^{t} .$ {q_k}(n,t) = \left({\matrix{{t + \left\lfloor {{{n - 1 - 2t} \over t}} \right\rfloor} \cr t \cr}} \right){2^t}.

Let us note that quasi (kA₁, 2A₂, 2A₃)-edge colouring by monochromatic paths of the graph 𝒫_n, such that exactly t monochromatic paths are coloured by A₂ or A₃, exists if and only if one of the numbers $\frac{n - 1 - 2 t}{k}$ {{n - 1 - 2t} \over k} or $\frac{n - 2 - 2 t}{k}$ {{n - 2 - 2t} \over k} is nonnegative integer. In order to deduce a direct formula for the (2, k)-distance Jacobsthal numbers of the second kind we introduce the following new notation: $T_{k}^{n} = t \in N \cup {0} : \frac{n - 1 - 2 t}{k} \in N \cup {0} or \frac{n - 2 - 2 t}{k} \in N \cup {0}\} .$ T_k^n = \left\{{t \in N \cup \{0\} :{{n - 1 - 2t} \over k} \in N \cup \{0\} \,{\rm{or}}\,{{n - 2 - 2t} \over k} \in N \cup \{0\}} \right\}. Observe that for example $T_{7}^{20} = 2, 6, 9\}$ T_7^{20} = \left\{{2,6,9} \right\} and $T_{1}^{n} = T_{2}^{n} = 0, 1, 2, \dots, \frac{n - 1}{2}⌋\}$ T_1^n = T_2^n = \left\{{0,1,2,\; \ldots ,\left\lfloor {{{n - 1} \over 2}} \right\rfloor} \right\} for all positive integers n. Moreover, $T_{k}^{n} \neq \emptyset$ T_k^n \ne \emptyset for all positive integers k and n ≥ 2.

Now we can state the analogue of Theorem 7 for the numbers J⁽²⁾(k, n).

Theorem 9

Let k ≥ 1, n ≥ 2 and $0 \leq t \leq \frac{n - 1}{2}⌋$ 0 \le t \le \left\lfloor {{{n - 1} \over 2}} \right\rfloor be integers. Then $J^{(2)} (k, n) = \sum_{t \in T_{k}^{n}} (\begin{matrix} t + \frac{n - 1 - 2 t}{k}⌋ \\ t \end{matrix}) 2^{t} .$ {J^{\left(2 \right)}}\left({k,n} \right) = \sum\limits_{t \in T_k^n} {\left({\matrix{{t + \left\lfloor {{{n - 1 - 2t} \over k}} \right\rfloor} \cr t \cr}} \right)} {2^t}.

Proof

As an immediate consequence of the graph interpretation of the numbers J⁽²⁾(k, n), given in Theorem 3, we obtain the following equality for k ≥ 1 and n ≥ 2: $J^{(2)} (k, n) = \sum_{t \in T_{k}^{n}} q_{k} (k, n) .$ {J^{\left(2 \right)}}\left({k,n} \right) = \sum\limits_{t \in T_k^n} {{q_k}\left({k,n} \right)} .

Corollary 4 makes it obvious that the desired formula holds.

3.

Identities

In this part we give the identities for the (2, k)-distance Jacobsthal numbers. For special values of parameters we can reduce them to the well-known identities for the classical Jacobsthal numbers.

Theorem 10

Let k ≥ 1, m ≥ 1, n ≥ 0 be integers and let n + k − 2 ≥ 0. Then for i = 1, 2 we have

(i)
$J^{(i)} (k, n) + 2 \sum_{i = 1}^{m} J^{(i)} (k, n - 2 + ik) = J^{(i)} (k, n + mk)$ {J^{\left(i \right)}}\left({k,n} \right) + 2\sum\limits_{i = 1}^m {{J^{\left(i \right)}}\left({k,n - 2 + ik} \right) = {J^{\left(i \right)}}\left({k,n + mk} \right)} ,
(ii)
$2^{m} J^{(i)} (k, n) + \sum_{i = 1}^{m} 2^{m - i} J^{(i)} (k, n - k + 2 i) = J^{(i)} (k, n + 2 m)$ {2^m}{J^{\left(i \right)}}\left({k,n} \right) + \sum\limits_{i = 1}^m {{2^{m - i}}{J^{\left(i \right)}}\left({k,n - k + 2i} \right) = {J^{\left(i \right)}}\left({k,n + 2m} \right)} .

Proof

In the proof of the part (i) we use the induction on m. If m = 1 then the equation is obvious. Assume that formula in (i) is true for an arbitrary m ≥ 1. We will prove that $J^{(i)} (k, n) + 2 \sum_{i = 1}^{m} J^{(i)} (k, n - 2 + ik) = J^{(i)} (k, n + (m + 1) k) .$ {J^{\left(i \right)}}\left({k,n} \right) + 2\sum\limits_{i = 1}^m {{J^{\left(i \right)}}\left({k,n - 2 + ik} \right) = {J^{\left(i \right)}}\left({k,n + (m + 1)k} \right)} .

By the induction hypothesis and the definition of J⁽ⁱ⁾(k, n), we have $\begin{array}{l} J^{(i)} (k, n) + 2 \sum_{i = 1}^{m + 1} J^{(i)} (k, n - 2 + ik) \\ = J^{(i)} (k, n) + 2 \sum_{i = 1}^{m} J^{(i)} (k, n - 2 + ik) + 2 J^{(i)} (k, n - 2 + (m + 1) k) \\ = J^{(i)} (k, n + mk) + 2 J^{(i)} (k, n - 2 + (m + 1) k) = J^{(i)} (k, n + (m + 1) k), \end{array}$ \matrix{{{J^{\left(i \right)}}\left({k,n} \right) + 2\sum\limits_{i = 1}^{m + 1} {{J^{\left(i \right)}}\left({k,n - 2 + ik} \right)}} \hfill \cr {\,\,\,\,\,\, = {J^{\left(i \right)}}\left({k,n} \right) + 2\sum\limits_{i = 1}^m {{J^{\left(i \right)}}\left({k,n - 2 + ik} \right) + 2{J^{\left(i \right)}}\left({k,n - 2 + \left({m + 1} \right)k} \right)}} \hfill \cr {\,\,\,\,\,\, = {J^{\left(i \right)}}\left({k,n + mk} \right) + 2{J^{\left(i \right)}}\left({k,n - 2 + \left({m + 1} \right)k} \right) = {J^{\left(i \right)}}\left({k,n + \left({m + 1} \right)k} \right),} \hfill \cr} which ends the proof of (i). The identity (ii) can be proved by the same method.

Putting n = 1 in identity (i) of Theorem 10 we obtain

Corollary 5

Let k, n be positive integers and let i = 1, 2. Then $1 + 2 \sum_{i = 1}^{m} J^{(i)} (k, ik - 1) = J^{(i)} (k, mk + 1) .$ 1 + 2\sum\limits_{i = 1}^m {{J^{\left(i \right)}}\left({k,ik - 1} \right) = {J^{\left(i \right)}}\left({k,mk + 1} \right).}

For k = 1 we obtain the well-known identity for the classical Jacobsthal numbers: $1 + 2 \sum_{i = 1}^{m} J_{i - 1} = J_{m + 1} .$ 1 + 2\sum\limits_{i = 1}^m {{J_{i - 1}} = {J_{m + 1}}.}

Putting k = 1 and n = 0 or n = 1 respectively in Theorem 10, we obtain the well-known identities for the classical Jacobsthal numbers $\sum_{i = 1}^{m} 2^{m - i} J_{2 i - 1} = J_{2 m}, 2^{m} + \sum_{i = 1}^{m} 2^{m - i} J_{2 i} = J_{2 m + 1} .$ \sum\limits_{i = 1}^m {{2^{m - i}}{J_{2i - 1}} = {J_{2m}}} ,\,\,\,\,\,\,{2^m} + \sum\limits_{i = 1}^m {{2^{m - i}}{J_{2i}} = {J_{2m + 1}}} .

We can use the (kA₁, 2A₂, 2A₃)-edge colouring of the path 𝒫_n to obtain the following identity for the (2, k)-distance Jacobsthal numbers of the first kind.

Theorem 11

Let k ≥ 1, m ≥ k and n ≥ k be integers. Then $\begin{matrix} J^{(1)} (k, m + n) = 2 J^{(1)} (k, m) J^{(1)} (k, n - 1) + 2 J^{(1)} (k, m - 1) J^{(1)} (k, n) \\ + \sum_{i = 1}^{k} J^{(1)} (k, m + 1 - i) J^{(1)} (k, n - k + i) . \end{matrix}$ \matrix{{{J^{\left(1 \right)}}\left({k,m + n} \right) = 2{J^{\left(1 \right)}}\left({k,m} \right){J^{\left(1 \right)}}\left({k,n - 1} \right) + 2{J^{\left(1 \right)}}\left({k,m - 1} \right){J^{\left(1 \right)}}\left({k,n} \right)} \cr {+ \sum\limits_{i = 1}^k {{J^{\left(1 \right)}}\left({k,m + 1 - i} \right){J^{\left(1 \right)}}\left({k,n - k + i} \right).}} \cr}

Proof

Consider the (kA₁, 2A₂, 2A₃)-edge colouring by monochromatic paths of the graph 𝒫_m+n. Let $V (P_{m + n}) = x_{1}, x_{2}, \dots, x_{m}, x_{m + 1}, \dots, x_{m + n}\}$ V\left({{{\mathcal P}_{m + n}}} \right) = \left\{{{x_1},{x_2}, \ldots ,{x_m},{x_{m + 1}}, \ldots ,{x_{m + n}}} \right\} be the set of vertices of this graph with the numbering in the natural fashion. Then $E (P_{m + n}) = x_{1} x_{2}, x_{2} x_{3}, \dots, x_{m} x_{m + 1}, \dots, x_{m + n - 1} x_{m + n}\} .$ E\left({{{\mathcal P}_{m + n}}} \right) = \left\{{{x_1}{x_2},{x_2}{x_3}, \ldots ,{x_m}{x_{m + 1}}, \ldots ,{x_{m + n - 1}}{x_{m + n}}} \right\}. By ρ(k, m + n) we denote the number of all (kA₁, 2A₂, 2A₃)-edge colourings by monochromatic paths of the graph 𝒫_m+n. Let ρ_A₁ (k, m + n), ρ_A₂ (k, m + n) and ρ_A₃ (k, m + n) denote the number of (kA₁, 2A₂, 2A₃)-edge colourings by monochromatic paths of the graph 𝒫_m+n, with A₁(x_mx_m₊₁), A₂(x_mx_m₊₁) and A₃(x_mx_m₊₁), respectively. In the case A₁(x_mx_m₊₁) we have k possibilities of the position in the graph 𝒫_m+n of the A₁-monochromatic path that includes an edge x_mx_m₊₁. It may be each of the following paths $\begin{matrix} x_{m + 1 - i} x_{m + 2 - i} \dots x_{m + k + 1 - i}, & i = 0, 1, \dots k . \end{matrix}$ \matrix{{{x_{m + 1 - i}}{x_{m + 2 - i}} \ldots {x_{m + k + 1 - i}},\;} & {i = 0,1,\; \ldots k.} \cr} For a fixed i = 0, 1, . . . k, the number of all (kA₁, 2A₂, 2A₃)-edge colourings by monochromatic paths of the graph 𝒫_m+n equals to the product of the number of all (kA₁, 2A₂, 2A₃)-edge colourings by monochromatic paths of the graph 𝒫_m+1−i and the number of all (kA₁, 2A₂, 2A₃)-edge colourings by monochromatic paths of the graph 𝒫_n−k+i. Therefore ρ_A₁ (k, m + n) is equal to the sum of this products. In both cases A₂(x_mx_m₊₁) or A₃(x_mx_m₊₁) there are two possibilities of the position in the graph 𝒫_m+n of the A_i-monochromatic path, i = 1, 2, that includes an edge x_mx_m₊₁. It can be any of the paths: x_m₋₁x_mx_m₊₁ or x_mx_m₊₁x_m₊₂. Consequently ρ_A₂ (k, m + n) and ρ_A₃ (k, m+n) both are equal to the number of all (kA₁, 2A₂, 2A₃)-edge colourings by monochromatic paths of the graph 𝒫_m multiplied by the number of all (kA₁, 2A₂, 2A₃)-edge colourings by monochromatic paths of the graph 𝒫_n−1 plus the number of all (kA₁, 2A₂, 2A₃)-edge colourings by monochromatic paths of the graph 𝒫_m−1 multiplied by the number of all (kA₁, 2A₂, 2A₃)-edge colourings by monochromatic paths of the graph 𝒫_n.

Thus from Theorem 3 we have $\begin{array}{l} ρ_{A_{2}} (k, m + n) & = ρ_{A_{3}} (k, m + n) \\ = J^{(1)} (k, m) J^{(1)} (k, n - 1) + J^{(1)} (k, m - 1) J^{(1)} (k, n) \end{array}$ \matrix{{{\rho_{{A_2}}}\left({k,m + n} \right)} \hfill & {= {\rho_{{A_3}}}\left({k,m + n} \right)} \hfill \cr {} \hfill & {= {J^{\left(1 \right)}}\left({k,m} \right){J^{\left(1 \right)}}\left({k,n - 1} \right) + {J^{\left(1 \right)}}\left({k,m - 1} \right){J^{\left(1 \right)}}\left({k,n} \right)} \hfill \cr} and $ρ_{A_{1}} (k, m + n) = \sum_{i = 1}^{k} J^{(1)} (k, m + 1 - i) J^{(1)} (k, n - k + i) .$ {\rho_{{A_1}}}\left({k,m + n} \right) = \sum\limits_{i = 1}^k {{J^{\left(1 \right)}}\left({k,m + 1 - i} \right){J^{\left(1 \right)}}\left({k,n - k + i} \right)} . Since ρ(k, m + n) = J⁽¹⁾(k, m + n) and $ρ (k, m + n) = ρ_{A_{1}} (k, m + n) + ρ_{A_{2}} (k, m + n) + ρ_{A_{3}} (k, m + n),$ \rho \left({k,m + n} \right) = {\rho_{{A_1}}}\left({k,m + n} \right) + {\rho_{{A_2}}}\left({k,m + n} \right) + {\rho_{{A_3}}}\left({k,m + n} \right), then the assertion follows.

Putting k = 1 in Theorem 11 we obtain the well-known formula for the classical Jacobsthal numbers J_m+n = 2J_mJ_n−1 + 2J_m−1J_n + J_mJ_n = J_mJ_n+1 + 2J_m−1J_n.

The following identity for the (2, k)-distance Jacobsthal numbers of the second kind can be deduced by Theorem 2 and Theorem 11.

Corollary 6

Let k ≥ 1, m ≥ k and n ≥ k be integers. Then $\begin{array}{l} J^{(2)} (k, m + n) & = 2 (\sum_{j = 0}^{m - 1} {(- 1)}^{j} J^{(2)} (k, m - 1 - j)) J^{(2)} (k, n) \\ + 2 (\sum_{j = 0}^{m - 1} {(- 1)}^{j} J^{(2)} (k, m - j)) J^{(2)} (k, n - 1) \\ + \sum_{i = 1}^{k} (\sum_{j = 0}^{m + 1 - i} {(- 1)}^{j} J^{(2)} (k, m + 1 - i - j)) J^{(2)} (k, n - k + i) . \end{array}$ \matrix{{{J^{\left(2 \right)}}\left({k,m + n} \right)} \hfill & {= 2\left({\sum\limits_{j = 0}^{m - 1} {{{(- 1)}^j}{J^{\left(2 \right)}}\left({k,m - 1 - j} \right)}} \right){J^{\left(2 \right)}}\left({k,n} \right)} \hfill \cr {} \hfill & {\,\,\,\, + 2\left({\sum\limits_{j = 0}^{m - 1} {{{(- 1)}^j}{J^{\left(2 \right)}}\left({k,m - j} \right)}} \right){J^{\left(2 \right)}}\left({k,n - 1} \right)} \hfill \cr {} \hfill & {+ \mathop \sum \nolimits_{i = 1}^k (\sum\limits_{j = 0}^{m + 1 - i} {{{(- 1)}^j}{J^{\left(2 \right)}}\left({k,m + 1 - i - j} \right)){J^{\left(2 \right)}}\left({k,n - k + i} \right).}} \hfill \cr}

4.

Matrix representations

Matrix representations of sequences give the possibility of deducing some properties of the terms of these sequences. For matrix generators of the Fibonacci numbers and the like, see [7] and [11].

In this section, we give matrix representations for the (2, k)-distance Jacobsthal numbers. Using these representations we obtain among other things Cassini-like formulas and some interesting identities for these numbers.

At the beginning, basing on the method used in [3], we introduce the matrix generator M_k for the numbers J⁽ⁱ⁾(k, n), i = 1, 2 where k ≥ 2. Let us recall the recurrence relation for these numbers: $J^{(i)} (k, n) = J^{(i)} (k, n - k) + 2 J^{(i)} (k, n - 2) for n \geq k + 1 .$ {J^{\left(i \right)}}\left({k,n} \right) = {J^{\left(i \right)}}\left({k,n - k} \right) + 2{J^{\left(i \right)}}\left({k,n - 2} \right)\,\,\,{\rm{for}}\,\,\,n \ge k + 1.

Let for a positive integer k ≥ 2, M_k be the matrix of the form [m_lj]_k×k where for a fixed 1 ≤ j ≤ k, a number m_1j is the coefficient of J⁽ⁱ⁾(k, n − j) in the above recurrence formula. Moreover, for 2 ≤ s ≤ k we have $m_{sj} = \begin{array}{l} 1 & if j = l - 1, \\ 0 & otherwise . \end{array}$ {m_{sj}} = \left\{{\matrix{1 \hfill & {{\rm{if}}\;j = l - 1,} \hfill \cr 0 \hfill & {{\rm{otherwise}}.} \hfill \cr}} \right.

According to this definition we obtain the following matrices for k = 2, 3, 4, . . . $\begin{matrix} M_{2} = \begin{array}{l} 0 & 3 \\ 1 & 0 \end{array}], & M_{3} = \begin{array}{l} 0 & 2 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}], & M_{4} = \begin{array}{l} 0 & 2 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}], \dots, \end{matrix}$ \matrix{{{M_2} = \left[ {\matrix{0 \hfill & 3 \hfill \cr 1 \hfill & 0 \hfill \cr}} \right],} & {{M_3} = \left[ {\matrix{0 \hfill & 2 \hfill & 1 \hfill \cr 1 \hfill & 0 \hfill & 0 \hfill \cr 0 \hfill & 1 \hfill & 0 \hfill \cr}} \right],} & {{M_4} = \left[ {\matrix{0 \hfill & 2 \hfill & 0 \hfill & 1 \hfill \cr 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \cr 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill \cr 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill \cr}} \right], \ldots ,} \cr} and in general $M_{k} = \begin{array}{l} 0 & 2 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}] .$ {M_k} = \left[ {\matrix{0 \hfill & 2 \hfill & 0 \hfill & 0 \hfill & 1 \hfill \cr 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \cr 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \cr 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \cr 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill \cr}} \right].

Now, for a fixed integer k ≥ 2, we introduce the matrix $A_{k}^{(i)}$ A_k^{\left(i \right)} of initial conditions. It is the following matrix of order k: $A_{k}^{(i)} = \begin{matrix} J^{(i)} (k, 2 k - 2) & J^{(i)} (k, 2 k - 3) & \dots & J^{(i)} (k, k) & J^{(i)} (k, k - 1) \\ J^{(i)} (k, 2 k - 3) & J^{(i)} (k, 2 k - 4) & \dots & J^{(i)} (k, k - 1) & J^{(i)} (k, k - 2) \\ ⋮ & ⋮ & ⋮ & ⋮ \\ J^{(i)} (k, k) & J^{(i)} (k, k - 1) & \dots & J^{(i)} (k, 2) & J^{(i)} (k, 1) \\ J^{(i)} (k, k - 1) & J^{(i)} (k, k - 2) & \dots & J^{(i)} (k, 1) & J^{(i)} (k, 0) \end{matrix}],$ A_k^{\left(i \right)} = \left[ {\matrix{{{J^{\left(i \right)}}\left({k,2k - 2} \right)} & {{J^{\left(i \right)}}\left({k,2k - 3} \right)} & \ldots & {{J^{\left(i \right)}}\left({k,k} \right)} & {{J^{\left(i \right)}}\left({k,k - 1} \right)} \cr {{J^{\left(i \right)}}\left({k,2k - 3} \right)} & {{J^{\left(i \right)}}\left({k,2k - 4} \right)} & \ldots & {{J^{\left(i \right)}}\left({k,k - 1} \right)} & {{J^{\left(i \right)}}\left({k,k - 2} \right)} \cr \vdots & \vdots & {} & \vdots & \vdots \cr {{J^{\left(i \right)}}\left({k,k} \right)} & {{J^{\left(i \right)}}\left({k,k - 1} \right)} & \ldots & {{J^{\left(i \right)}}\left({k,2} \right)} & {{J^{\left(i \right)}}\left({k,1} \right)} \cr {{J^{\left(i \right)}}\left({k,k - 1} \right)} & {{J^{\left(i \right)}}\left({k,k - 2} \right)} & \ldots & {{J^{\left(i \right)}}\left({k,1} \right)} & {{J^{\left(i \right)}}\left({k,0} \right)} \cr}} \right], where i = 1, 2, k ≥ 2.

Using the same method as in [3], we obtain the following result.

Theorem 12

Let k ≥ 2, n ≥ 1 be integers. Then for i = 1, 2 we have ${(M_{k})}^{n} \cdot A_{k}^{(i)} = \begin{matrix} J^{(i)} (k, n + 2 k - 2) & J^{(i)} (k, n + 2 k - 3) & \dots & J^{(i)} (k, n + k) & J^{(i)} (k, n + k - 1) \\ J^{(i)} (k, n + 2 k - 3) & J^{(i)} (k, n + 2 k - 4) & \dots & J^{(i)} (k, n + k - 1) & J^{(i)} (k, n + k - 2) \\ ⋮ & ⋮ & ⋮ & ⋮ \\ J^{(i)} (k, n + k) & J^{(i)} (k, n + k - 1) & \dots & J^{(i)} (k, n + 2) & J^{(i)} (k, n + 1) \\ J^{(i)} (k, n + k - 1) & J^{(i)} (k, n + k - 2) & \dots & J^{(i)} (k, n + 1) & J^{(i)} (k, n) \end{matrix}] .$ {({M_k})^n} \cdot A_k^{\left(i \right)} = \left[ {\matrix{{{J^{\left(i \right)}}\left({k,n + 2k - 2} \right)} & {{J^{\left(i \right)}}\left({k,n + 2k - 3} \right)} & \ldots & {{J^{\left(i \right)}}\left({k,n + k} \right)} & {{J^{\left(i \right)}}\left({k,n + k - 1} \right)} \cr {{J^{\left(i \right)}}\left({k,n + 2k - 3} \right)} & {{J^{\left(i \right)}}\left({k,n + 2k - 4} \right)} & \ldots & {{J^{\left(i \right)}}\left({k,n + k - 1} \right)} & {{J^{\left(i \right)}}\left({k,n + k - 2} \right)} \cr \vdots & \vdots & {} & \vdots & \vdots \cr {{J^{\left(i \right)}}\left({k,n + k} \right)} & {{J^{\left(i \right)}}\left({k,n + k - 1} \right)} & \ldots & {{J^{\left(i \right)}}\left({k,n + 2} \right)} & {{J^{\left(i \right)}}\left({k,n + 1} \right)} \cr {{J^{\left(i \right)}}\left({k,n + k - 1} \right)} & {{J^{\left(i \right)}}\left({k,n + k - 2} \right)} & \ldots & {{J^{\left(i \right)}}\left({k,n + 1} \right)} & {{J^{\left(i \right)}}\left({k,n} \right)} \cr}} \right].

Two next theorems will be helpful in formulating Cassini-like formulas for the (2, k)-distance Jacobsthal numbers J⁽ⁱ⁾(k, n).

Theorem 13

For all integers k ≥ 2 the following equality holds $det M_{k} = \begin{array}{l} - 3 & for k = 2, \\ {(- 1)}^{k + 1} & for k \geq 3 . \end{array}$ \det \;{M_k} = \left\{{\matrix{{- 3} \hfill & {for\;k = 2,} \hfill \cr {{{(- 1)}^{k + 1}}} \hfill & {for\;k \ge 3.} \hfill \cr}} \right.

Proof

It is easy to see that $M_{2} = \begin{matrix} 0 & 3 \\ 1 & 0 \end{matrix}| = - 3$ {M_2} = \left| {\matrix{0 & 3 \cr 1 & 0 \cr}} \right| = - 3 . For k ≥ 3 we calculate the determinant |M_k| using the Laplace expansion by the last column of the matrix M_k. This expansion gives $det M_{k} = \begin{matrix} 0 & 2 & 0 & \dots & 0 & 1 \\ 1 & 0 & 0 & \dots & 0 & 0 \\ 0 & 1 & 0 & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & \dots & 0 & 0 \\ 0 & 0 & 0 & \dots & 1 & 0 \end{matrix}| = {(- 1)}^{k + 1} \cdot \begin{array}{l} 1 & 0 & \dots & 0 \\ 0 & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 \dots & 1 \end{array}| = {(- 1)}^{k + 1} .$ \det \;{M_k} = \left| {\matrix{0 & 2 & 0 & \ldots & 0 & 1 \cr 1 & 0 & 0 & \ldots & 0 & 0 \cr 0 & 1 & 0 & \ldots & 0 & 0 \cr \vdots & \vdots & \vdots & {} & \vdots & \vdots \cr 0 & 0 & 0 & \ldots & 0 & 0 \cr 0 & 0 & 0 & \ldots & 1 & 0 \cr}} \right| = {(- 1)^{k + 1}} \cdot \left| {\matrix{1 \hfill & 0 \hfill & \ldots \hfill & 0 \hfill \cr 0 \hfill & 1 \hfill & \ldots \hfill & 0 \hfill \cr \vdots \hfill & \vdots \hfill & {} \hfill & \vdots \hfill \cr 0 \hfill & 0 \hfill & {0 \ldots} \hfill & 1 \hfill \cr}} \right| = {(- 1)^{k + 1}}.

Thus the theorem is proved.

Theorem 14

Let k ≥ 2 be an integer. Then (3) $det A_{k}^{(1)} = {(- 1)}^{\frac{k (k - 1)}{2}}$ \det \;A_k^{\left(1 \right)} = {(- 1)^{{{k\left({k - 1} \right)} \over 2}}} and (4) $det A_{k}^{(2)} = \begin{array}{l} - 1 & for k = 2, \\ 0 & for odd k, \\ 2 (1) \frac{1}{2} k & for even k > 2 . \end{array}$ \det \;A_k^{\left(2 \right)} = \left\{{\matrix{{- 1} \hfill & {for\;k = 2,} \hfill \cr 0 \hfill & {for\;odd\;k,} \hfill \cr {2\left(1 \right)\;{1 \over 2}k} \hfill & {for\;even\;k > 2.} \hfill \cr}} \right.

Proof

Let k ≥ 2 be an integer. In the proof of equality (3) the auxiliary sequence $J_{B}^{(1)} (k, n)$ J_B^{\left(1 \right)}\left({k,n} \right) will be very helpful. We define it as follows $J_{B}^{(1)} (k, 0) = J_{B}^{(1)} (k, 1) = \dots = J_{B}^{(1)} (k, k - 2) = 0, J_{B}^{(1)} (k, k - 1) = 1$ J_B^{\left(1 \right)}\left({k,0} \right) = J_B^{\left(1 \right)}\left({k,1} \right) = \cdots = J_B^{\left(1 \right)}\left({k,k - 2} \right) = 0,\,\,\,\,\,\,\,J_B^{\left(1 \right)}\left({k,k - 1} \right) = 1 and (5) $J_{B}^{(1)} (k, n) = J_{B}^{(1)} (k, n - 2) + 2 J_{B}^{(1)} (k, n - k) for n \geq k .$ J_B^{\left(1 \right)}\left({k,n} \right) = J_B^{\left(1 \right)}\left({k,n - 2} \right) + 2J_B^{\left(1 \right)}\left({k,n - k} \right)\,\,\,\,\,{\rm for}\,\,\,\,\,n \ge k. Now we define a matrix $B_{k}^{(1)}$ B_k^{\left(1 \right)} of order k, k ≥ 2, whose elements are the terms of the sequence $J_{B}^{(1)} (k, n)$ J_B^{\left(1 \right)}\left({k,\;n} \right) : $B_{k}^{(1)} = \begin{matrix} J_{B}^{(1)} (k, 2 k - 2) & J_{B}^{(1)} (k, 2 k - 3) & \dots & J_{B}^{(1)} (k, k) & J_{B}^{(1)} (k, k - 1) \\ J_{B}^{(1)} (k, 2 k - 3) & J_{B}^{(1)} (k, 2 k - 4) & \dots & J_{B}^{(1)} (k, k - 1) & J_{B}^{(1)} (k, k - 2) \\ ⋮ & ⋮ & ⋮ & ⋮ \\ J_{B}^{(1)} (k, k) & J_{B}^{(1)} (k, k - 1) & \dots & J_{B}^{(1)} (k, 2) & J_{B}^{(1)} (k, 1) \\ J_{B}^{(1)} (k, k - 1) & J_{B}^{(1)} (k, k - 2) & \dots & J_{B}^{(1)} (k, 1) & J_{B}^{(1)} (k, 0) \end{matrix}] .$ B_k^{\left(1 \right)} = \left[ {\matrix{{J_B^{\left(1 \right)}\left({k,2k - 2} \right)} & {J_B^{\left(1 \right)}\left({k,2k - 3} \right)} & \ldots & {J_B^{\left(1 \right)}\left({k,k} \right)} & {J_B^{\left(1 \right)}\left({k,k - 1} \right)} \cr {J_B^{\left(1 \right)}\left({k,2k - 3} \right)} & {J_B^{\left(1 \right)}\left({k,2k - 4} \right)} & \ldots & {J_B^{\left(1 \right)}\left({k,k - 1} \right)} & {J_B^{\left(1 \right)}\left({k,k - 2} \right)} \cr \vdots & \vdots & {} & \vdots & \vdots \cr {J_B^{\left(1 \right)}\left({k,k} \right)} & {J_B^{\left(1 \right)}\left({k,k - 1} \right)} & \ldots & {J_B^{\left(1 \right)}\left({k,2} \right)} & {J_B^{\left(1 \right)}\left({k,1} \right)} \cr {J_B^{\left(1 \right)}\left({k,k - 1} \right)} & {J_B^{\left(1 \right)}\left({k,k - 2} \right)} & \ldots & {J_B^{\left(1 \right)}\left({k,1} \right)} & {J_B^{\left(1 \right)}\left({k,0} \right)} \cr}} \right]. From the definition of the sequence $J_{B}^{(1)} (k, n)$ J_B^{\left(1 \right)}\left({k,\;n} \right) it follows that $B_{k}^{(1)} = \begin{matrix} J_{B}^{(1)} (k, 2 k - 2) & J_{B}^{(1)} (k, 2 k - 3) & \dots & J_{B}^{(1)} (k, k) & 1 \\ J_{B}^{(1)} (k, 2 k - 3) & J_{B}^{(1)} (k, 2 k - 4) & \dots & 1 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ \\ J_{B}^{(1)} (k, k) & 1 & \dots & 0 & 0 \\ 1 & 0 & \dots & 0 & 0 \end{matrix}] .$ B_k^{\left(1 \right)} = \left[ {\matrix{{J_B^{\left(1 \right)}\left({k,2k - 2} \right)} & {J_B^{\left(1 \right)}\left({k,2k - 3} \right)} & \ldots & {J_B^{\left(1 \right)}\left({k,k} \right)} & 1 \cr {J_B^{\left(1 \right)}\left({k,2k - 3} \right)} & {J_B^{\left(1 \right)}\left({k,2k - 4} \right)} & \ldots & 1 & 0 \cr \vdots & \vdots & {} & \vdots & \vdots \cr {J_B^{\left(1 \right)}\left({k,k} \right)} & 1 & \ldots & 0 & 0 \cr 1 & 0 & \ldots & 0 & 0 \cr}} \right]. Using k − 1 times the Laplace expansion by the last column we can calculate the determinant of the matrix $B_{k}^{(1)}$ B_k^{\left(1 \right)} as follows $\begin{array}{l} det B_{k}^{(1)} & = {(- 1)}^{k + 1} \cdot {(- 1)}^{k} \dots \cdot \cdot {(- 1)}^{3} \\ = {(- 1)}^{k - 1} \cdot {(- 1)}^{k - 2} \dots \cdot \cdot (1) \\ = {(- 1)}^{1 + 2 + \dots + (k - 1)} = {(- 1)}^{\frac{k (k - 1)}{2}} . \end{array}$ \matrix{{{\det}\,B_k^{\left(1 \right)}} \hfill & {= {{(- 1)}^{k + 1}} \cdot {{(- 1)}^k} \cdots \cdot \cdot {{(- 1)}^3}} \hfill \cr {} \hfill & {= {{(- 1)}^{k - 1}} \cdot {{(- 1)}^{k - 2}} \cdots \cdot \cdot \left(1 \right)} \hfill \cr {} \hfill & {= {{(- 1)}^{1 + 2 + \cdots + \left({k - 1} \right)}} = {{(- 1)}^{{{k\left({k - 1} \right)} \over 2}}}.} \hfill \cr} Using the recurrence formula (5) and definitions of matrices $A_{k}^{(1)}$ A_k^{\left(1 \right)} and $B_{k}^{(1)}$ B_k^{\left(1 \right)} one can prove that for k ≥ 2 the equality $A_{k}^{(1)} = B_{k}^{(1)} {(M_{k}^{T})}^{k - 2}$ A_k^{\left(1 \right)} = B_k^{\left(1 \right)}{(M_k^T)^{k - 2}} holds where $M_{k}^{T}$ M_k^T denotes the transpose of the matrix M_k. Therefore by properties of determinants we obtain $det A_{k}^{(1)} = det B_{k}^{(1)} \cdot {(det (M_{k}^{T}))}^{k - 2} .$ \det A_k^{\left(1 \right)} = \det B_k^{\left(1 \right)} \cdot {(\det \;(M_k^T))^{k - 2}}.

Consequently, for k = 2 we have $det A_{k}^{(1)} = - 1$ {\det}A_k^{\left(1 \right)} = - 1 and for k ≥ 2 by Theorem 13 we get $det A_{k}^{(1)} = {(- 1)}^{\frac{k (k - 1)}{2}} \cdot {(- 1)}^{(k + 1) (k - 2)} .$ \det A_k^{\left(1 \right)} = {(- 1)^{{{k\left({k - 1} \right)} \over 2}}} \cdot {(- 1)^{\left({k + 1} \right)\left({k - 2} \right)}}. Note that the expression (k + 1)(k − 2) is even for all integers k ≥ 3, hence $det A_{k}^{(1)} = {(- 1)}^{\frac{k (k - 1)}{2}}$ \det \,A_k^{\left(1 \right)} = {(- 1)^{{{k\left({k - 1} \right)} \over 2}}} which completes the proof of (3). For the proof of the equalities (4) we define a new sequence $J_{B}^{(2)} (k, n)$ J_B^{\left(2 \right)}\left({k,n} \right) : (6) $\begin{array}{l} J_{B}^{(2)} (k, 0) = J_{B}^{(2)} (k, k - 1) = 1, \\ J_{B}^{(2)} (k, 1) = J_{B}^{(2)} (k, 2) = \dots = J_{B}^{(2)} (k, k - 2) = 0, \\ J_{B}^{(2)} (k, n) = J_{B}^{(2)} (k, n - k) + 2 J_{B}^{(2)} (k, n - 2) for k > 2, n \geq k, \end{array}$ \matrix{{J_B^{\left(2 \right)}\left({k,0} \right) = J_B^{\left(2 \right)}\left({k,k - 1} \right) = 1,} \hfill \cr {J_B^{\left(2 \right)}\left({k,1} \right) = J_B^{\left(2 \right)}\left({k,2} \right) = \cdots = J_B^{\left(2 \right)}\left({k,k - 2} \right) = 0,} \hfill \cr {J_B^{\left(2 \right)}\left({k,n} \right) = J_B^{\left(2 \right)}\left({k,n - k} \right) + 2J_B^{\left(2 \right)}\left({k,n - 2} \right)\,\,\,\,\,{\rm{for}}\,\,\,\,k > 2,\,\,n \ge k,} \hfill \cr} and an auxiliary matrix $B_{k}^{(2)}$ B_k^{\left(2 \right)} of order k: $\begin{array}{l} B_{k}^{(2)} & = [\begin{matrix} J_{B}^{(2)} (k, 2 k - 2) & J_{B}^{(2)} (k, 2 k - 3) & \dots & J_{B}^{(2)} (k, k) & J_{B}^{(2)} (k, k - 1) \\ J_{B}^{(2)} (k, 2 k - 3) & J_{B}^{(2)} (k, 2 k - 4) & \dots & J_{B}^{(2)} (k, k - 1) & J_{B}^{(2)} (k, k - 2) \\ ⋮ & ⋮ & ⋮ & ⋮ \\ J_{B}^{(2)} (k, k) & J_{B}^{(2)} (k, k - 1) & \dots & J_{B}^{(2)} (k, 2) & J_{B}^{(2)} (k, 1) \\ J_{B}^{(2)} (k, k - 1) & J_{B}^{(2)} (k, k - 2) & \dots & J_{B}^{(2)} (k, 1) & J_{B}^{(2)} (k, 0) \end{matrix}] \\ = [\begin{matrix} J_{B}^{(2)} (k, 2 k - 2) & J_{B}^{(2)} (k, 2 k - 3) & \dots & J_{B}^{(2)} (k, k) & 1 \\ J_{B}^{(2)} (k, 2 k - 3) & J_{B}^{(2)} (k, 2 k - 4) & \dots & 1 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ \\ J_{B}^{(2)} (k, k) & 1 & \dots & 0 & 0 \\ 1 & 0 & \dots & 0 & 1 \end{matrix}] . \end{array}$ \matrix{{B_k^{\left(2 \right)}} \hfill & {= \left[ {\matrix{{J_B^{\left(2 \right)}\left({k,2k - 2} \right)} & {J_B^{\left(2 \right)}\left({k,2k - 3} \right)} & \ldots & {J_B^{\left(2 \right)}\left({k,k} \right)} & {J_B^{\left(2 \right)}\left({k,k - 1} \right)} \cr {J_B^{\left(2 \right)}\left({k,2k - 3} \right)} & {J_B^{\left(2 \right)}\left({k,2k - 4} \right)} & \ldots & {J_B^{\left(2 \right)}\left({k,k - 1} \right)} & {J_B^{\left(2 \right)}\left({k,k - 2} \right)} \cr \vdots & \vdots & {} & \vdots & \vdots \cr {J_B^{\left(2 \right)}\left({k,k} \right)} & {J_B^{\left(2 \right)}\left({k,k - 1} \right)} & \ldots & {J_B^{\left(2 \right)}\left({k,2} \right)} & {J_B^{\left(2 \right)}\left({k,1} \right)} \cr {J_B^{\left(2 \right)}\left({k,k - 1} \right)} & {J_B^{\left(2 \right)}\left({k,k - 2} \right)} & \ldots & {J_B^{\left(2 \right)}\left({k,1} \right)} & {J_B^{\left(2 \right)}\left({k,0} \right)} \cr}} \right]} \hfill \cr {} \hfill & {= \left[ {\matrix{{J_B^{\left(2 \right)}\left({k,2k - 2} \right)} & {J_B^{\left(2 \right)}\left({k,2k - 3} \right)} & \ldots & {J_B^{\left(2 \right)}\left({k,k} \right)} & 1 \cr {J_B^{\left(2 \right)}\left({k,2k - 3} \right)} & {J_B^{\left(2 \right)}\left({k,2k - 4} \right)} & \ldots & 1 & 0 \cr \vdots & \vdots & {} & \vdots & \vdots \cr {J_B^{\left(2 \right)}\left({k,k} \right)} & 1 & \ldots & 0 & 0 \cr 1 & 0 & \ldots & 0 & 1 \cr}} \right].} \hfill \cr} By definitions of matrices $A_{k}^{(2)}$ A_k^{\left(2 \right)} and $B_{k}^{(2)}$ B_k^{\left(2 \right)} and by the recurrence formula (6) we can deduce the following relationship between $A_{k}^{(2)}$ A_k^{\left(2 \right)} and $B_{k}^{(2)}$ B_k^{\left(2 \right)} (7) $A_{k}^{(2)} = B_{k}^{(2)} {(M_{k}^{T})}^{k - 2}$ A_k^{\left(2 \right)} = B_k^{\left(2 \right)}{(M_k^T)^{k - 2}} Using basic properties of determinants one can prove that $det B_{k}^{(2)} = \begin{array}{l} 0 & for odd k, \\ 2 {(- 1)}^{\frac{1}{2} k} & for even k . \end{array}$ \det B_k^{\left(2 \right)} = \left\{{\matrix{0 \hfill & {{\rm{for}}\;{\rm{odd}}\;k,} \hfill \cr {2{{(- 1)}^{{1 \over 2}k}}} \hfill & {{\rm{for}}\;{\rm{even}}\;k.} \hfill \cr}} \right. From this and from the formula (7) it follows immediately that $det A_{k}^{(2)} = 0$ \det A_k^{\left(2 \right)} = 0 for odd k. For even k > 2 by applying Theorem 13 we get $det A_{k}^{(2)} = 2 {(- 1)}^{\frac{1}{2} k} {(- 1)}^{(k + 1) (k - 2)} = 2 {(- 1)}^{\frac{1}{2} k} .$ \det A_k^{\left(2 \right)} = 2{(- 1)^{{1 \over 2}k}}{(- 1)^{\left({k + 1} \right)\left({k - 2} \right)}} = 2{(- 1)^{{1 \over 2}k}}. Moreover we can see that for k = 2 we have $det A_{k}^{(2)} = \begin{array}{l} 1 & 1 \\ 1 & 0 \end{array}] = - 1$ \det A_k^{\left(2 \right)} = \left[ {\matrix{1 \hfill & 1 \hfill \cr 1 \hfill & 0 \hfill \cr}} \right] = - 1 , thus the proof is completed.

As a consequence of Theorem 13 and Theorem 14 we obtain Cassini-like formulas for the (2, k)-distance Jacobsthal numbers.

Corollary 7

Let k ≥ 2, n ≥ 2 be integers and let i = 1, 2. Then

(i)
$det {(M_{k})}^{n} \cdot A_{k}^{(1)}] = \begin{array}{l} {(- 1)}^{n + 1} 3^{n} & for k = 2, \\ {(- 1)}^{n (k + 1) + \frac{k (k - 1)}{2}} & for k \geq 3, \end{array}$ \det \left[ {{{({M_k})}^n} \cdot A_k^{\left(1 \right)}} \right] = \left\{{\matrix{{{{(- 1)}^{n + 1}}{3^n}} \hfill & {for\;k = 2,} \hfill \cr {{{(- 1)}^{n\left({k + 1} \right) + {{k\left({k - 1} \right)} \over 2}}}} \hfill & {for\;k \ge 3,} \hfill \cr}} \right.
(ii)
$det [{(M_{k})}^{n} \cdot A_{k}^{(2)}] = \begin{array}{l} - {(- 3)}^{n} & for k = 2, \\ 0 & for odd k, \\ 2 (1)^{\frac{3}{2} k + 1} & for even k > 2 . \end{array}$ \det [{({M_k})^n} \cdot A_k^{\left(2 \right)}] = \left\{{\matrix{{- {{(- 3)}^n}} \hfill & {for\;k = 2,} \hfill \cr 0 \hfill & {for\;odd\;k,} \hfill \cr {2\left(1 \right){\;^{{3 \over 2}k + 1}}} \hfill & {for\;even\;k > 2.} \hfill \cr}} \right.

Theorem 15

Let k ≥ 3 and n ≥ 2k − 4 be integers. Then (M_k)ⁿ is of the form $\begin{matrix} J^{(1)} (k, n + 1) & J^{(1)} (k, n + 2) & J^{(1)} (k, n - k + 3) & \dots & J^{(1)} (k, n) \\ J^{(1)} (k, n) & J^{(1)} (k, n + 1) & J^{(1)} (k, n - k + 2) & \dots & J^{(i)} (k, n - 1) \\ ⋮ & ⋮ & ⋮ & ⋮ \\ J^{(1)} (k, n - k + 3) & J^{(1)} (k, n - k + 4) & J^{(1)} (k, n - 2 k + 5) & \dots & J^{(i)} (k, n - k + 2) \\ J^{(1)} (k, n - k + 2) & J^{(1)} (k, n - k + 3) & J^{(1)} (k, n - 2 k + 4) & \dots & J^{(i)} (k, n - k + 1) \end{matrix}] .$ \left[ {\matrix{{{J^{(1)}}\left({k,n + 1} \right)} & {{J^{(1)}}\left({k,n + 2} \right)} & {{J^{(1)}}\left({k,n - k + 3} \right)} & \ldots & {{J^{(1)}}\left({k,n} \right)} \cr {{J^{(1)}}\left({k,n} \right)} & {{J^{(1)}}\left({k,n + 1} \right)} & {{J^{(1)}}\left({k,n - k + 2} \right)} & \ldots & {{J^{(i)}}\left({k,n - 1} \right)} \cr \vdots & \vdots & \vdots & {} & \vdots \cr {{J^{(1)}}\left({k,n - k + 3} \right)} & {{J^{(1)}}\left({k,n - k + 4} \right)} & {{J^{(1)}}\left({k,n - 2k + 5} \right)} & \ldots & {{J^{(i)}}\left({k,n - k + 2} \right)} \cr {{J^{(1)}}\left({k,n - k + 2} \right)} & {{J^{(1)}}\left({k,n - k + 3} \right)} & {{J^{(1)}}\left({k,n - 2k + 4} \right)} & \ldots & {{J^{(i)}}\left({k,n - k + 1} \right)} \cr}} \right].

Proof

(By induction on n.) Let k ≥ 3 be a fixed integer. For n = 2k − 4 we can check the equation by inspection. Assume that the equation is true for all integers 2k − 3, 2k − 2, . . . , n. To show that it is true also for n + 1 it is enough to use the induction hypothesis, definition of J⁽¹⁾(k, n) and the equation (M_k)ⁿ⁺¹ = (M_k)ⁿM_k.

By Theorem 13 and Theorem 15 we obtain new Cassini-like formulas for the (2, k)-distance Jacobsthal numbers of the first kind J⁽¹⁾(k, n).

Corollary 8

For all positive integers k, n, we have $det {(M_{k})}^{n} = \begin{array}{l} {(- 3)}^{n} & if k = 2 and n \geq 1, \\ {(- 1)}^{n (k + 1)} & if k \geq 3 and n \geq 2 k - 4 . \end{array}$ \det {({M_k})^n} = \left\{{\matrix{{{{(- 3)}^n}} \hfill & {if\;k = 2\;and\;n \ge 1,} \hfill \cr {{{(- 1)}^{n\left({k + 1} \right)}}} \hfill & {if\;k \ge 3\;and\;n \ge 2k - 4.} \hfill \cr}} \right.

Note that from Corollary 8 it follows that for all integers n ≥ 2k − 4 the determinant of the matrix (M_k)ⁿ can be expressed as follows (8) $det {(M_{k})}^{n} = \begin{array}{l} {(- 1)}^{n} & if k is even, \\ 1 & if k is odd . \end{array}$ \det {({M_k})^n} = \left\{{\matrix{{{{(- 1)}^n}} \hfill & {{\rm{if}}\;k\;{\rm{is\;even}},} \hfill \cr 1 \hfill & {{\rm{if}}\;k\;{\rm{is\;odd}}.} \hfill \cr}} \right. For example putting k = 3 in (8) we obtain the following identity for the numbers J⁽¹⁾(k, n).

Corollary 9

For every integer n ≥ 2 we have $\begin{array}{l} {(J^{(1)} (3, n + 1))}^{2} J^{(1)} (3, n - 2) + {(J^{(1)} (3, n))}^{3} \\ + {(J^{(1)} (3, n - 1))}^{2} J^{(1)} (3, n + 2) - 2 J^{(1)} (3, n - 1) J^{(1)} (3, n) J^{(1)} (3, n + 1) \\ - J^{(1)} (3, n - 2) J^{(1)} (3, n) J^{(1)} (3, n + 2) = 1 . \end{array}$ \matrix{{{{({J^{\left(1 \right)}}(3,n + 1))}^2}{J^{\left(1 \right)}}(3,n - 2) + {{({J^{\left(1 \right)}}(3,n))}^3}} \hfill \cr {+ {{({J^{\left(1 \right)}}\left({3,n - 1} \right))}^2}{J^{\left(1 \right)}}\left({3,n + 2} \right) - 2{J^{\left(1 \right)}}\left({3,n - 1} \right){J^{\left(1 \right)}}\left({3,n} \right){J^{\left(1 \right)}}\left({3,n + 1} \right)} \hfill \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cr {- {J^{\left(1 \right)}}\left({3,n - 2} \right){J^{\left(1 \right)}}\left({3,n} \right){J^{\left(1 \right)}}\left({3,n + 2} \right) = 1.}}

5.

Concluding remarks

The interpretation of the (2, k)-distance Jacobsthal numbers with respect to the number of (kA₁, 2A₂, 2A₃)-edge colourings by monochromatic paths of some graphs gives the motivation for studying different kinds of (a₁A₁, a₂A₂, a₃A₃)-edge colourings of special graphs. For an arbitrary positive integer k some interesting results connected with the number of (A₁, A₂, kA₃)-edge colourings, ((k − 1)A₁, (k − 1)A₂, kA₃)-edge colourings and (kA₁, kA₂, 2A₃)-edge colourings of some trees are recently obtained in [12],[13],[14] and [17].

On (k1A1, k2A2, k3A3)-Edge Colourings in Graphs and Generalized Jacobsthal Numbers

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