Tribonacci Sequences With Certain Indices And Their Sums

In this paper, we derive new recurrence relations and generating matrices for the sums of usual Tribonacci numbers and 4n subscripted Tribonacci sequences, fT4ng ; and their sums. We obtain explicit formulas and combinatorial representations for the sums of terms of these sequences. Finally we represent relationships between these sequences and permanents of certain matrices. 1. Introduction The Tribonacci sequence is de…ned by for n > 1 Tn+1 = Tn + Tn 1 + Tn 2 where T0 = 0; T1 = 1; T2 = 1: The few …rst terms are 0; 1; 1; 2; 4; 7; 13; 24; 44; 81; 149; : : : : We de…ne Tn = 0 for all n 0: The Tribonacci sequence is a well known generalization of the Fibonacci sequence. In (see page 527-536, [3]), one can …nd some known properties of Tribonacci numbers. For example, the generating matrix of fTng is given by Q = 24 1 1 1 1 0 0 0 1 0 35n = 24 Tn+1 Tn + Tn 1 Tn Tn Tn 1 + Tn 2 Tn 1 Tn 1 Tn 2 + Tn 3 Tn 2 35 : For further properties of Tribonacci numbers, we refer to [1, 4, 5]. Let Sn = Pn k=0 Tk: (1.1) In this paper, we obtain generating matrices for the sequences fTng,fT4ng, fSng and fS4ng : (The second result follows from a third order recurrence for T4n:) We also obtain Binet-type explicit and closed-form formulas for Sn and S4n: Further on, we present relationships between permanents of certain matrices and all the above-mentioned sequences. 2000 Mathematics Subject Classi…cation. 11B37, 15A36, 11P.