More Identities On The Tribonacci Numbers
نویسنده
چکیده
In this paper, we use a simple method to derive di¤erent recurrence relations on the Tribonacci numbers and their sums. By using the companion matrices and generating matrices, we get more identities on the Tribonacci numbers and their sums, which are more general than that given in literature [E. Kilic, Tribonacci Sequences with Certain Indices and Their Sum, Ars Combinatoria 86 (2008), 13-22.]. 1. Introduction The Tribonacci sequence is like the Fibonacci sequence, but instead of starting with two predetermined terms, the sequence starts with three predetermined terms and each term afterwards is the sum of the preceding three terms, that is Tn = Tn 1 + Tn 2 + Tn 3, n 3 (1.1) where T0 = T1 = 0; T2 = 1: The rst few tribonacci numbers are: 0; 0; 1; 1; 2; 4; 7; 13; 24; 44; 81; 149; 274; 504; 927; 1705; 3136; 5768; The tribonacci constant 1+ 3 p 19+3 p 33+ 3 p 19 3 p 33 3 is the ratio toward which adjacent tribonacci numbers tend. It is a root of the polynomial x x x 1, approximately 1.83929 , and also satis es the equation x 2x+1 = 0. In [1], the author derives new recurrence relations and generating matrices for the sums of usual Tribonacci numbers fSng and 4n subscripted Tribonacci numbers fT4ng, and their sums fS4ng, where Sn = Pn k=0 Tk. In this paper, we intend to give the more identities on the Tribonacci numbers fTn+wg, arbitrary subscripted Tribonacci numbers fTw(n+h)g, and their sums fSn+wg; fSw(n+h)g, where w and h are arbitrary positive integers. 2. Another Recurrence Relation By the recurrence (1.1), we have two expressions: Tn = Tn 1 + Tn 2 + Tn 3, and Tn 1 = Tn 2+Tn 3+Tn 4, subtract the second expression from 2000 Mathematics Subject Classi cation. 11B37, 15A36.
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عنوان ژورنال:
- Ars Comb.
دوره 100 شماره
صفحات -
تاریخ انتشار 2011