THE GENERALIZED TRIBONACCI NUMBERS WITH NEGATIVE SUBSCRIPTS

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1 #A3 INTEGERS 14 (014) THE GENERALIZED TRIBONACCI NUMBERS WITH NEGATIVE SUBSCRIPTS Kantaphon Kuhapatanakul 1 Dept. of Mathematics, Faculty of Science, Kasetsart University, Bangkok, Thailand fscikpkk@ku.ac.th Lalitphat Sukruan Dept. of Mathematics, Rajamangala University of Technology Suvarnabhumi Suphanburi, Thailand brk1 5@hotmail.com Received: 8/1/13, Revised: /5/14, Accepted: 4/19/14, Published: 7/10/14 Abstract Generalized tribonacci numbers R n are defined through the recurrence R n+1 ar n + br n 1 + cr n. A generating matrix of three tribonacci sequences with negative subscripts is defined and used to establish identities connecting these sequences which is analogous to the matrix of Shannon and Horadam. We derive an explicit formula for the generalized tribonacci numbers with negative subscripts. 1. Introduction Let L(a, b, c) be the set of all third-order recurrent sequences {R n } nz satisfying the relation R n ar n 1 + br n + cr n 3, where a, b, c are positive integers. Three generalized tribonacci sequences {J n }, {K n } and {L n } in L(a, b, c) are uniquely determined by taking special values n 0, 1,, namely, (i) J 0 0, J 1 1 and J a, (ii) K 0 1, K 1 0 and K b, (iii) L 0 0, L 1 0 and L c. 1 Supported by the Kasetsart University Research and Development Institute (KURDI), Thailand.

2 INTEGERS: 14 (014) These sequences have been studied by many researchers (for more details see [5]- [7]). If a b c 1, then {J n } is a sequence of the classical tribonacci numbers, say {T n }. It can be written R n as a linear combination of J n, K n and L n, namely, R n+1 R J n + R 1 K n + R 0 L n, and the following relations are easily proved, see [4], [5] or [7], J n+1 aj n + K n, K n+1 bj n + cj n 1, and L n+1 cj n. (1) In 197, Shannon and Horadam [7] constructed the 3 3 matrix and computed the n th power of this matrix 4 a b c 3n J 3 n+ K n+ L n+ J n+1 K n+1 L n J n K n L n and they showed that (a + b + c 1) n J i J n+3 + (1 a)j n+ + (1 a b)j n+1 1, J n+1 [n/] [n/3] j0 i + j n i j j i + j a n i 3j b i c j ; see also [1], [], [5], [6]. The sequences {J n }, {K n } and {L n } can be defined for negative values of n by using the definition of any recurrent relation and initial conditions. The first few terms of them are shown in the following table. n J n K n L n /c a/c b/c 3 b/c (ab + c)/c (b ac)/c 4 (b ac)/c 3 (a c ab bc)/c 3 (c + abc b 3 )/c 3 In this article we construct certain matrices for J n, K n, L n and partial sums of J n to derive interesting identities involving these numbers. We also derive an expansion of J n in a partial sum of binomial coe cients.. Matrix Representations For n N, define the 3 3 matrices A and C n as follows 3 b/c a/c 1/c A and C n 4 L 3 n 1 K n 1 J n 1 L n K n J n L n+1 K n+1 J n+1

3 INTEGERS: 14 (014) 3 Theorem 1. For all n N, we have A n C n. Proof. (Induction on n) Using the above table (section 1), one can see that A 1 C 1. Assume A n C n holds for n > 1. By our assumption and a matrix multiplication, we get A n+1 A n A C n A, which, by using all equations in (1), satisfies A n+1 C n A C n+1. Thus, complete the proof. Since A n+m A n A m, equating the (, 1), (, ) and (, 3)-entries on both sides of this matrix equation, we get the following corollary. Corollary 1. For m, n N, we get the relation where R n is J n, K n or L n. R m n L m R n 1 + K m R n + J m R n+1, Taking n and m in Corollary 1, we get R m R 3 L m + R K m + R 1 J m, R n 1 c ( br n 1 ar n + R n+1 ), respectively. Next, for n N, we define the 4 4 matrices B and D n as follows B 61/c A 5 and D n 6S n S n C n 5, 0 where S n P n J i and S 0 0. Theorem. For all n N, we have B n D n. S n+1 Proof. Since L n+1 cj n, we can write S n 1 S n + 1 c L n. Combining the above identity and the result of Theorem 1, we write D n D n 1 B. By using induction on n, the result still holds. Corollary. For all positive integers m, n, we have S m n 1 S m 1 + L m 1 S n 1 + K m 1 S n + J m 1 S n+1. () Proof. Since D m+n D m D n, we have the equation () by equating the (, 1)-entry on both sides of this matrix equation. Now we derive an explicit formula for partial sums of J n. We have the following theorem.

4 INTEGERS: 14 (014) 4 Theorem 3. For n 1, we have n 1 J i a + b + c 1 (1 cj n 1 + (a 1)J n J n+1 ). (3) Proof. Put m 1 in the equation () of Corollary, we get This equation equivalent to cs n 1 as n bs n 1 + S n+1. (a + b + c 1)S n+1 1 cj n + J n 1 ( b c) + J n ( a b c). By the definition of J n, we can rewrite the last equation to obtain (3). Taking a b c 1 in identity (3), we obtain n T i 1 (1 T n 1 T n+1 ). 3. Expansions Definition 1. Let n, i be non-negative integers with n B(n, i) b(n+i)/3c j0 i n j i j i. Denote a i j b n i j c j. It is easy to see that we can write B(n, i) in the form of the recursive recurrence as B(n, i) bb(n 1, i) + ab(n 1, i 1) + cb(n, i 1). (4) Theorem 4. For non-negative integer n, we have Proof. We see that J n bn/c ( 1) n i B(n i, i). (5) cn i+1 J c 1, J 3 bc and J 4 (b ac)c 3.

5 INTEGERS: 14 (014) 5 By induction on n, assume that identity (5) holds for all n 0, 1,,..., k the definition of J k, the identity (4) and the inductive hypothesis, we get 1. By cj k J k+1 aj k bj k 1 so 8 >: 3)/c b 1)/c ( 1) k 3 i c k i B(k i 3, i) a 1)/c c k i B(k i 1, i) )/c ( 1) k i bk/c c k i 1 B(k i, i 1) a ( 1) k 1 b c k B(k 1, 0) b ( 1) >< k B(k, 0) c k bk/c ( 1) k c k B(k, 0) 1)/c 1)/c ( 1) k i c k i B(k i, i), J k 1)/c ( 1) k i c k i 1 B(k i, i) c k i B(k i 1, i 1) c k i B(k i 1, i) c k i B(k i, i) ; k is odd k/ a c k i B(k i, i) + ; k is even c bk/c ( 1) k i B(k i, i). ck i+1 Showing that (5) holds for n k, thereby proving the theorem. We can rewrite (5) in terms of binomial coe cients by using Definition 1. Corollary 3. For non-negative integer n, we have bn/c bn/3c i n i j J n ( 1) n i a i j b n i j c i+j n 1. j i j0 Taking a b c 1, we get an explicit formula for the tribonacci numbers, so bn/c bn/3c i n i j T n ( 1) n i. (6) j i j0 Acknowledgment The authors would like to thank the anonymous referee for a number of helpful suggestions.

6 INTEGERS: 14 (014) 6 References [1] V.E. Hoggatt, Jr. and M Bicknell, Generalized Fibonacci polynomials, Fibonacci Quart. 11(5) (1973), [] E. Kilic, Tribonacci sequences with certain indices and their sums. Ars Combinatoria 86 (008), 13. [3] P.Y. Lin, De Moivre-type identities for the Tribonacci numbers, Fibonacci Quart. 6() (1988), [4] S. Pethe, Some identities for tribonacci sequences. Fibonacci Quart. 6() (1988), [5] S. Rabinowitz, Algorithmic manipulation of third-order linear recurrences. Fibonacci Quart. 34(5) (1996), [6] A.G. Shannon, Iterative formulas associated with generalized third-order recurrence relations. SIAM J. Appl. Math. 3(3) (197), [7] A.G. Shannon and A.F. Horadam, Some properties of third-order recurrence relations. Fibonacci Quart. 10() (197),