On Generalized Fibonacci Numbers 1

arXiv:1503.05305v1 [math.NT] 18 Mar 2015 Applied Mathematical Sciences, Vol. x, 2015, no. xx, xxx - xxx HIKARI Ltd, www.m-hikari.com On Generalized Fibonacci Numbers1 Jerico B. Bacani∗ and Julius Fergy T. Rabago† Department of Mathematics and Computer Science College of Science University of the Philippines Baguio Baguio City 2600, Philippines ∗ [email protected], † [email protected] Copyright c 2015 Jerico B. Bacani and Julius Fergy T. Rabago. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We provide a formula for the nth term of the k-generalized Fibonaccilike number sequence using the k-generalized Fibonacci number or knacci number, and by utilizing the newly derived formula, we show that the limit of the ratio of successive terms of the sequence tends to a root of the equation x+x−k = 2. We then extend our results to k-generalized Horadam (kGH) and k-generalized Horadam-like (kGHL) numbers. In dealing with the limit of the ratio of successive terms of kGH and kGHL, a lemma due to Z. Wu and H. Zhang [8] shall be employed. Finally, we remark that an analogue result for k-periodic k-nary Fibonacci sequence can also be derived. Mathematics Subject Classification: 11B39, 11B50. Keywords: k-generalized Fibonacci numbers, k-generalized Fibonacci-like numbers, k-generalized Horadam numbers, k-generalized Horadam-like numbers, convergence of sequences 1 to appear by April 2015 2 1 J. B. Bacani and J. F. T. Rabago Introduction A well-known recurrence sequence of order two is the widely studied Fibonacci sequence {Fn }∞ n=1 , which is defined recursively by the recurrence relation F1 = F2 = 1, Fn+1 = Fn + Fn−1 (n ≥ 1). (1) Here, it is conventional to define F0 = 0. In the past decades, many authors have extensively studied the Fibonacci sequence and its various generalizations (cf. [2, 3, 4, 6, 7]). We want to contribute more in this topic, so we present our results on the k-generalized Fibonacci numbers or k-nacci numbers and of its some generalizations. In particular, we derive a formula for the k-generalized Fibonacci-like sequence using k-nacci numbers. Our work is motivated by the following statement: Consider the set of sequences satisfying the relation Sn = Sn−1 + Sn−2 . Since the sequence {Sn } is closed under term-wise addition (resp. multiplication) by a constant, it can be viewed as a vector space. Any such sequence is uniquely determined by a choice of two elements, so the vector space is two-dimensional. If we denote such sequence as (S0 , S1 ), then the Fibonacci sequence Fn = (0, 1) and the shifted Fibonacci sequence Fn−1 = (1, 0) are seen to form a canonical basis for this space, yielding the identity: Sn = S1 Fn + S0 Fn−1 (2) for all such sequences {Sn }. For example, if S is the Lucas sequence 2, 1, 3, 4, 7, . . ., then we obtain Sn : = Ln = 2Fn−1 + Fn . One of our goals in this paper is to find an analogous result of the equation (2) for k-generalized Fibonacci numbers. The result is significant because it provides an explicit formula for the nth term of a k-nacci-like (resp. kgeneralized Horadam and k-generalized Horadam-like) sequences without the need of solving a system of equations. By utilizing the formula, we also show that the limit of the ratio of successive terms of a k-nacci sequence tends to a root of the equation x + x−k = 2. We then extend our results to k-generalized Horadam and k-generalized Horadam-like sequences. We also remark that an analogue result for k-periodic k-nary Fibonacci sequences can be derived. 2 Fibonacci-like sequences of higher order We start off this section with the following definition. Definition 2.1. Let n ∈ N ∪ {0} and k ∈ N\{1}. Consider the sequences (k) (k) ∞ {Fn }∞ n=0 and {Gn }n=0 having the following properties: 3 On Generalized Fibonacci Numbers Fn(k) and   0, 1, =  Pk 0 ≤ n < k − 1; n = k − 1; (k) i=1 Fn−i , n > k − 1, G(k) n = ( (k) (k) Gn , Pk (k) i=1 Gn−i , 0 ≤ n ≤ k − 1; n > k − 1, (k) (3) (4) (k) and Gn 6= 0 for some n ∈ [0, k−1]. The terms Fn and Gn satisfying (3) and (4) are called the nth k-generalized Fibonacci number or nth k-step Fibonacci number (cf. [7]), and nth k-generalized Fibonacci-like number, respectively. (k) For {Fn }∞ n=0 , some famous sequences of this type are the following: k 2 3 4 5 name of sequence first few terms of the sequence Fibonacci 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, . . . Tribonacci 0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, . . . Tetranacci 0, 0, 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, . . . Pentanacci 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 31, 61, 120, . . . (k) (k) ∞ By considering the sequences {Fn }∞ n=0 and {Gn }n=0 we obtain the following relation. (k) (k) Theorem 2.2. Let Fn and Gn be the nth k-generalized Fibonacci and kgeneralized Fibonacci-like numbers, respectively. Then, for all natural numbers n ≥ k, ! m+1 k−3 X X (k) (k) (k) (k) (k) (5) Gm+1 Fn−1−j + Gk−1 Fn(k) . G(k) n = G0 Fn−1 + m=0 j=0 Proof. We prove this using induction on n. Let k be fixed. Equation (5) is obviously valid for n < k. Now, suppose (5) is true for n ≥ r ≥ k where r ∈ N. Then, (k) Gr+1 = k X (k) G(r+1)−i i=1 k X (k) = G0 (k) F(r+1)−i−1 + k−3 X (k) Gm+1 m=0 i=1 (k) + Gk−1 k X j=0 (k) F(r+1)−i i=1 = (k) (k) G0 F(r+1)−1 + k−3 X m=0 m+1 X (k) Gm+1 m+1 X j=0 ! k X (k) F(r+1)−i−1−j i=1 (k) F(r+1)−1−j ! (k) !! (k) + Gk−1 Fr+1 . 4 J. B. Bacani and J. F. T. Rabago Remark 2.3. Using the formula obtained by G. P. B. Dresden (cf. [2, Theo(k) rem 1]), we can now express Gn explicitly in terms of n as follows: k X (k) G(k) n = G0 A(i; k)αin−2 + i=1 k−3 X m=0  G(k) m+1 m+1 k XX j=0 i=1  (k) A(i; k)αin−2−j +Gk−1 k X A(i; k)αin−1 , i=1 where A(i; k) = (αi − 1)[2 + (k + 1)(αi − 2)]−1 and α1 , α2 , . . . , αk are roots of (k) xk − xk−1 − · · · − 1 = 0. Another formula of Dresden for Fn (cf. [2, Theorem 2]) (k) can also be used to express Gn explicitly in terms of n. More precisely, we have   m+1 k k−3 k h i X X X X   (k) G(k) Round A(k)αin−2 + G(k) Round A(k)αin−2−j  n = G0 m+1 m=0 i=1 j=0 i=1 (k) + Gk−1 k X i=1   Round A(k)αin−1 , where A(k) = (α − 1)[2 + (k + 1)(α − 2)]−1 for all n ≥ 2 − k and for α the unique positive root of xk − xk−1 − · · · − 1 = 0. Extending to Horadam numbers In 1965, A. F. Horadam [5] defined a second-order linear recurrence sequence ∞ {Wn (a, b; p, q)}∞ n=0 , or simply {Wn }n=0 by the recurrence relation W0 = a, W1 = b, Wn+1 = pWn + qWn−1 , (n ≥ 2). The sequence generated is called the Horadam’s sequence which can be viewed easily as a certain generalization of {Fn }. The nth Horadam number Wn with initial conditions W0 = 0 and W1 = 1 can be represented by the following Binet’s formula: αn − β n Wn (0, 1; p, q) = (n ≥ 2), α−β 2 where α p and β are the roots of the pquadratic equation x − px − q = 0, i.e. 2 2 α = (p+ p + 4q)/2 and β = (p− p + 4q). We extend this definition to the concept of k-generalized Fibonacci sequence and we define the k-generalized Horadam (resp. Horadam-like) sequence as follows: Definition 2.4. Let qi ∈ N for i ∈ {1, 2, . . . , n}. For n ≥ k, the nth k(k) generalized Horadam sequence, denoted by {Un (0, . . . , 1; q1 , . . . , qk )}∞ n=0 , or (k) ∞ th simply {Un }n=0 , is a sequence whose n term is obtained by the recurrence relation k X (k) (k) (k) (k) (k) Un = q1 Un−1 + q2 Un−2 + · · · + qk Un−k = qi Un−i , (6) i=1 5 On Generalized Fibonacci Numbers (k) (k) with initial conditions Ui = 0 for all 0 ≤ i < k − 1 and Uk−1 = 1. Similarly, (k) the k-generalized Horadam-like sequence, denoted by {Vn (a0 , . . . , ak−1 ; q1 , . . . , qk )}∞ n=0 (k) , has the same recurrence relation given by equation (6) but or simply {Vn }∞ n=0 (k) with initial conditions Vi = ai for all 0 ≤ i ≤ k − 1 where ai ′ s ∈ N ∪ {0} with at least one of them is not zero. (k) It is easy to see that when q1 = · · · = qk = 1, then Un (0, . . . , 1; 1, . . . , 1) = (k) (k) (k) Fn and Vn (a0 , . . . , ak−1 ; 1, . . . , 1) = Gn . Using Definition 2.4 we obtain the following relation, which is an analogue of equation (5). (k) (k) Theorem 2.5. Let Un and Vn be the nth k-generalized Horadam and nth k-generalized Horadam-like numbers, respectively. Then, for all n ≥ k, (k) (k) Vn(k) = qk V0 Un−1 + k−3 X m=0 (k) Vm+1 m+1 X j=0 (k) qk−(m+1)+j Un−1−j ! (k) + Vk−1 Un(k) . (7) Proof. The proof uses mathematical induction and is similar to the proof of Theorem 2.2. Convergence properties In the succeeding discussions, we present the convergence properties of the (k) (k) ∞ (k) ∞ (k) ∞ sequences {Fn }∞ n=0 , {Gn }n=0 , {Un }n=0 , and {Vn }n=0 . First, it is known (k) (k) (e.g. in [7]) that limn→∞ Fn /F(n−1) = α, where α is a k-nacci constant. This constant is the unique positive real root of xk − xk−1 − · · · − 1 = 0 and can also be obtained by solving the zero of the polynomial xk (2 − x) − 1. Using this result, we obtain the following: Theorem 2.6. (k) lim G(k) n /Gn−1 = α, n→∞ where α the unique positive root of xk − xk−1 − · · · − 1 = 0. (8) 6 J. B. Bacani and J. F. T. Rabago (k) (k) Proof. The proof is straightforward. Letting n → ∞ in Gn /Gn−1 we have (k) lim G(k) n /Gn−1 n→∞   (k) (k) (k) + G F F n n−1−j k−1 m=0 j=0    = lim  P P (k) (k) (k) (k) (k) m+1 (k) n→∞ G0 Fn−2 + k−3 G F + G F m+1 n−2−j k−1 n−1 m=0 j=0     (k) Pk−3 (k) (k) Pm+1 Fn−1−j (k) Fn(k) + Gk−1 (k)  (k)  G0 + m=0 Gm+1 j=0 Fn−1 Fn−1     = lim  (k)  (k) Pk−3 n→∞ (k) (k) Pm+1 Fn−2−j (k) Fn−2 + Gk−1 G0 (k) + m=0 Gm+1 j=0 (k) Fn−1 Fn−1   Pk−3 (k) (k) P (k) −j G0 + m=0 Gm+1 m+1 + αGk−1 j=0 α   = P (k) (k) Pm+1 −(j+1) (k) α−1 G0 + k−3 G + Gk−1 α m+1 m=0 j=0  (k) (k) Pk−3  (k) (k) G0 Fn−1 + (k) Gm+1 Pm+1 = α. (k) (k) Now, to find the limit of Un /Un−1 (resp. Vn /Vn−1 ) as n → ∞ we need the following results due to Wu and Zhang [8]. Here, it is assumed that the qi ′ s satisfy the inequality qi ≥ qj ≥ 1 for all j ≥ i, where 1 ≤ i, j ≤ k with 2 ≤ k ∈ N. Lemma 2.7. [8] Let q1 , q2 , . . . , qk be positive integers with q1 ≥ q2 ≥ · · · ≥ qk ≥ 1 and k ∈ N\{1}. Then, the polynomial f (x) = xk − q1 xk−1 − q2 xk−2 − · · · − qk−1 x − qk , (9) (i) has exactly one positive real zero α with q1 < α < q1 + 1; and (ii) its other k − 1 zeros lie within the unit circle in the complex plane. Lemma 2.8. [8] Let k ≥ 2 and let {un }∞ n=0 be an integer sequence satisfying the recurrence relation given by un = q1 un−1 + q2 un−2 + · · · + qk−1 un−k+1 + qk un−k , n > k, (10) where q1 , q2 , . . . , qk ∈ N with initial conditions ui ∈ N ∪ {0} for 0 ≤ i < k and at least one of them is not zero. Then, a formula for un may be given by un = cαn + O(d−n ) (n → ∞), where c > 0, d > 1, and q1 < α < q1 + 1 is the positive real zero of f (x). We now have the following results. (11) On Generalized Fibonacci Numbers 7 Theorem 2.9. Let {Un }∞ n=0 be the integer sequence satisfying the recurrence (k) relation (6) with initial conditions Ui = 0 for all 0 ≤ i < k − 1, 2 ≤ k ∈ N (k) and Uk−1 = 1 with q1 ≥ q2 ≥ · · · ≥ qk ≥ 1. Then, Un(k) = cαn + O(d−n ) (n → ∞), (12) where c > 0, d > 1, and α ∈ (q1 , q1 + 1) is the positive real zero of f (x). Furthermore, (k) lim Un(k) /Un−1 = α. (13) n→∞ Proof. Equation (12) follows directly from Lemmas 2.7 and 2.8. To obtain (k) (k) (13), we simply use (12) and take the limit of the ratio Un /Un−1 as n → ∞; that is, we have the following manipulation: (k) cαn + O(d−n ) n→∞ cαn−1 + O(d−(n−1) ) cα + limn→∞ (O(d−n )/αn−1 ) = c + limn→∞ (O(d−(n−1) )/αn−1 ) = α. lim Un(k) /Un−1 = lim n→∞ Consequently, we have the following corollary. Corollary 2.10. Let {Vn }∞ n=0 be an integer sequence satisfying (6) but with (k) initial conditions Vi = ai for all 0 ≤ i ≤ k−1 where ai ′ s ∈ N∪{0} with atleast one of them is not zero. Furthermore, assume that q1 ≥ q2 ≥ · · · ≥ qk ≥ 1, where 2 ≤ k ∈ N then (k) lim Vn(k) /Vn−1 = α, (14) n→∞ where q1 < α < q1 + 1 is the positive real zero of f (x). Proof. The proof uses Theorem 2.5 and the arguments used are similar to the proof of Theorem 2.6. Remark 2.11. Observe that when qi = 1 for all i = 0, 1, . . . , k in Corollary (k) (k) (2.10), then limn→∞ r : = limn→∞ Fn /Fn−1 = α, where 1 < α < 2. Indeed, the limit of the ratio r is 2 as n increases. k-Periodic Fibonacci Sequences In [3], M. Edson and O. Yayenie gave a generalization of Fibonacci sequence. (a,b) They called it generalized Fibonacci sequence {Fn }∞ n=0 which they defined 8 J. B. Bacani and J. F. T. Rabago it by using a non-linear recurrence relation depending on two real parameters (a, b). The sequence is defined recursively as (a,b) F0 = 0, (a,b) F1 = 1, Fn(a,b) = ( (a,b) (a,b) aFn−1 + Fn−2 , if n is even, (a,b) (a,b) bFn−1 + Fn−2 , if n is odd. (15) This generalization has its own Binet-like formula and satisfies identities that are analogous to the identities satisfied by the classical Fibonacci sequence (see [3]). A further generalization of this sequence, which is called k-periodic Fibonacci sequence has been presented by M. Edson, S. Lewis, and O. Yayenie in [4]. A related result concerning to two-periodic ternary sequence is presented in [1] by M. Alp, N. Irmak and L. Szalay. We expect that analogous results of (5), (7), and (12) can easily be found for these generalizations of Fibonacci sequence. For instance, if we alter the starting values of (15), say we start at two numbers A and B and preserve the recurrence relation in (15), then we obtain a sequence that we may call 2-periodic Fibonacci-like sequence, which is defined as follows: ( (a,b) (a,b) aGn−1 + Gn−2 , if n is even, (a,b) (a,b) G0 = A, G1 = B, Gn(a,b) = (16) (a,b) (a,b) bGn−1 + Gn−2 , if n is odd. (a,b) ∞ }n=0 The first few terms of {Fn (a,b) ∞ }n=0 and {Gn (a,b) are as follows: (a,b) n Fn Gn 0 0 A 1 B 1 a aB + A 2 3 ab + 1 (ab + 1)B + bA a2 b + 2a (a2 b + 2a)B + (ab + 1)A 4 a2 b2 + 3ab + 1 (a2 b2 + 3ab + 1)B + (ab2 + 2b)A 5 3 2 2 3 2 6 a b + 4a b + 3a (a b + 4a2 b + 3a)B + (a2 b2 + 3ab + 1)A 7 a3 b3 + 5a2 b2 + 6ab + 1 (a3 b3 + 5a2 b2 + 6ab + 1)B + (a2 b3 + 4ab2 + 3b)A (a,b) (a,b) Suprisingly, by looking at the table above, Gn can be obtained using Fn (b,a) and Fn . More precisely, we have the following result. (a,b) (a,b) Theorem 2.12. Let Fn and Gn be the nth terms of the sequences defined in (15) and (16), respectively. Then, for all n ∈ N, the following formula holds (a,b) Gn(a,b) = G1 (a,b) Fn(a,b) + G0 (b,a) Fn−1 . (17) 9 On Generalized Fibonacci Numbers Proof. The proof is by induction on n. Evidently, the formula holds for n = 0, 1, 2. We suppose that the formula also holds for some n ≥ 2. Hence, we have (a,b) (a,b) Fn−1 + G0 (a,b) Fn(a,b) + G0 Gn−1 = G1 Gn(a,b) = G1 (a,b) (a,b) Fn−2 , (b,a) (a,b) Fn−1 . (b,a) Suppose that n is even. (The case when n is odd can be proven similarly.) So we have (a,b) (a,b) Gn+1 = aGn(a,b) + Gn−1     (a,b) (a,b) (a,b) (b,a) (a,b) (a,b) (a,b) (b,a) = a G1 Fn + G0 Fn−1 + G1 Fn−1 + G0 Fn−2     (b,a) (b,a) (a,b) (a,b) (a,b) aFn−1 + Fn−2 aFn(a,b) + Fn−1 + G0 = G1 (a,b) = G1 (a,b) (a,b) Fn+1 + G0 Fn(b,a) , proving the theorem. (a,b) The sequence {Gn }∞ n=0 has already been studied in ([3], Section 4). The (a,b) (a,b) ∞ authors [3] have related the two sequences {Fn }∞ }n=0 using n=0 and {Gn the formula  n−2⌊n/2⌋ b (a,b) (a,b) (a,b) (a,b) (a,b) Gn = G1 F n + G0 Fn−1 . (18) a Notice that by simply comparing the two identities (17) and (18), we see that  n−2⌊n/2⌋ b (b,a) (a,b) Fn−1 = Fn−1 , ∀n ∈ N. a (a,b) (a,b) The convergence property of {Fn+1 /Fn }∞ n=0 has also been discussed in ([3], Remark 2). It was shown that, for a = b, we have √ (a,b) Fn+1 α a + a2 + 4 −→ = as n −→ ∞. (19) (a,b) a 2 Fn (a,b) (a,b) Using (17) and (19), we can also determine the limit of the sequence {Gn+1 /Gn as n tends to infinity, and for a = b, as follows: (a,b) lim n→∞ Gn+1 (a,b) Gn = lim n→∞ G1 (a,a) G1 Fn+1 + G0 (a,b) Fn G1 (a,a) = (a,b) G1 (a,b) (a,b) (a,b) Fn (a,b) Fn−1 + G0 (a,a) limn→∞ (a,a) + G0 Fn+1 (a,a) Fn (b,a) (b,a) (a,a) + G0 limn→∞ = lim (a,a) Fn−1 (a,a) Fn n→∞ (a,a) Fn+1 + G0 (a,a) Fn G1 G1 (a,a) = αa−1 G1 (a,a) G1 (a,a) (a,a) Fn (a,a) Fn−1 + G0 (a,a) + G0 (a,a) + aα−1 G0 (a,a) = α . a (a,a) (a,a) } 10 J. B. Bacani and J. F. T. Rabago (a,b) For the case a 6= b, the ratio of successive terms of {Fn However, it is easy to see that (a,b) F2n (a,b) F2n+1 α −→ , b (a,b) F2n−1 (a,b) F2n α −→ , a } does not converge. (a,b) Fn+2 and (a,b) Fn −→ α + 1, √ where α = (ab + a2 b2 + 4ab)/2 (cf. [3]). Knowing all these limits, we can in(a,b) (a,b) (a,b) (a,b) vestigate the convergence property of the sequences {G2n /G2n−1 }, {G2n+1 /G2n }, (a,b) (a,b) (a,b) (b,a) and {Gn+2 /Gn }. Notice that Fn = Fn for every n ∈ {1, 3, 5, . . .}. So (b,a) (a,b) G2n lim n→∞ G(a,b) 2n−1 = (a,b) (a,b) G1 F2n lim n→∞ G(a,b) F (a,b) 1 2n−1 (a,b) (b,a) + G0 F2n−1 (a,b) (b,a) + G0 F2n−2 (b,a) (a,b) F2n G1 = lim n→∞ (a,b) +G = lim n→∞ (a,b) + G0 (b,a) F2n−1 G1 (a,b) F2n G1 (a,b) (a,b) F2n−2 (a,b) 0 F2n−1 (a,b) (a,b) = αa−1 G1 (a,b) G1 + (a,b) (a,b) F2n−1 (a,b) G1 (a,b) (a,b) F2n−2 + G0 (a,b) + G0 (a,b) aα−1 G0 (a,b) + G0 = (a,b) (b,a) F2n−1 α . a (a,b) Similarly, it can be shown that G2n+1 /G2n → α/b and Gn+2 /Gn → α + 1 as n → ∞. The recurrence relations discussed above can easily be extended into subscripts with real numbers. For instance, consider the piecewise defined function (a,b) G⌊x⌋ : (a,b) G0 (a,b) = A, G1 (a,b) = B, G⌊x⌋ =  (a,b) (a,b)   aG⌊x⌋−1 + G⌊x⌋−2 , if ⌊x⌋ is even,   bG(a,b) + G(a,b) , ⌊x⌋−2 ⌊x⌋−1 (20) if ⌊x⌋ is odd. Obviously, the properties of (16) will be inherited by (20). For example, (a,b) (a,b) (.2,.3) suppose G0 = 2, G1 = 3, a = 0.2, and b = 0.3. Then, G⌊x⌋ = (.2,.3) G1 (.2,.3) F⌊x⌋ (.2,.3) + G0 (.3,.2) F⌊x⌋−1 . Also, (.2,.3) lim x→∞ G2⌊x⌋ (.2,.3) G2⌊x⌋−1 (.2,.3) = 1.3839, lim x→∞ G2⌊x⌋+1 (.2,.3) G2⌊x⌋ (.2,.3) = 0.921886, lim x→∞ G⌊x⌋+2 (.2,.3) G⌊x⌋ (.1,.1) = 1.276807. (.1,.1) If a = b = 0.1, then the ratio of successive terms of {Gn } with G0 =2 (.1,.1) and G1 = 3 converges to 1.05125. See Figure 1 for the plots of these limits. Now, we may take the generalized Fibonacci sequence (15) a bit further by considering a 3-periodic ternary recurrence sequence related to the usual 11 On Generalized Fibonacci Numbers Figure 1: G0 = 2, G1 = 3, a = 0.2, b = 0.3; G0 = 2, G1 = 3, a = 0.1, b = 0.1. Tribonacci sequence: (a,b,c) Tn(a,b,c) (a,b,c) (a,b,c) = 0, T1 = 0, T2 = 1, T  0 (a,b,c) (a,b,c) (a,b,c)   aTn−1 + Tn−2 + Tn−3 , if n ≡ 0 (mod 3), (a,b,c) (a,b,c) (a,b,c) = bTn−1 + Tn−2 + Tn−3 , if n ≡ 1 (mod 3), (n ≥ 3)   cT (a,b,c) + T (a,b,c) + T (a,b,c) , if n ≡ 2 (mod 3). n−1 n−2 n−3 (21) (a,b,c) Suppose we define the sequence {Un }∞ n=0 satisfying the same recurrence (a,b,c) (a,b,c) equation as in (21) but with arbitrary initial conditions U0 , U1 , and (a,b,c) (a,b,c) ∞ (a,b,c) ∞ U2 . Then, the sequences {Un }n=0 and {Tn }n=0 are related as follows:   (a,b,c) (a,b,c) (b,c,a) (c,a,b) (a,b,c) (b,c,a) (a,b,c) + U2 Tn , (n ≥ 2). Tn−1 + Tn−2 Un(a,b,c) = U0 Tn−1 + U1 (a1 ,a2 ,...,ak ) Remark 2.13. In general, the k-periodic k-nary sequence {Fn (k) {Fn } related to k-nacci sequence: (k) (k) F0 = F1       (k) Fn =      (k) }: = (k) = · · · = Fk−2 = 0, Fk−1 = 1, P (k) (k) a1 Fn−1 + kj=2 Fn−j , if n ≡ 0 (mod k), P (k) (k) a2 Fn−1 + kj=2 Fn−j , if n ≡ 1 (mod k), (n ≥ k) .. . P (k) (k) ak Fn−1 + kj=2 Fn−j , if n ≡ −1 (mod k). (22) 12 J. B. Bacani and J. F. T. Rabago (a ,a ,...,a ) (k) and the sequence {Gn 1 2 k } : = {Gn } defined in the same recurrence equa(k) (k) (k) tion (22) but with arbitrary initial conditions G0 , G1 , . . . , Gk−1 are related in the following fashion: ! m+1 k−3 X (k;j+1) X (k) (k) (k) (k;1) (23) Gm+1 Fn−1−j + Gk−1 F(k) G(k) n , n = G0 Fn−1 + m=0 j=0 where (k; j) : = (aj+1 , aj+2 , . . . , ak , a1 , a2 , . . . , aj ), j = 0, 1, . . . , k −1. This result can be proven by mathematical induction and we leave this to the interested reader. 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