Generating Functions of Binary Products of Tribonacci and Tribonacci Lucas Polynomials and Special Numbers

Tribonacci and Tribonacci-Lucas Numbers via the Determinants of Special Matrices

Incomplete generalized Tribonacci polynomials and numbers

The main object of this paper is to present a systematic investigation of a new class of polynomials – incomplete generalized Tribonacci polynomials and a class of numbers associated with the familiar Tribonacci polynomials. The various results obtained here for these classes of polynomials and numbers include explicit representations, generating functions, recurrence relations and summation fo...

Tribonacci Numbers and the Brocard - Ramanujan Equation

Let (Tn)n≥0 be the Tribonacci sequence defined by the recurrence Tn+2 = Tn+1 + Tn + Tn−1, with T0 = 0 and T1 = T2 = 1. In this short note, we prove that there are no integer solutions (u,m) to the Brocard-Ramanujan equation m! + 1 = u2 where u is a Tribonacci number.

On the sum of reciprocal Tribonacci numbers

In this paper we consider infinite sums derived from the reciprocals of the Fibonacci numbers, and infinite sums derived from the reciprocals of the square of the Fibonacci numbers. Applying the floor function to the reciprocals of these sums, we obtain equalities that involve the Fibonacci numbers.

A Tribonacci-Like Sequence of Composite Numbers

We find a new Tribonacci-like sequence of positive integers 〈x0, x1, x2, . . .〉 given by xn = xn−1 + xn−2 + xn−3 , n ≥ 3, and gcd(x0, x1, x2) = 1 that contains no prime numbers. We show that the sequence with initial values x0 = 151646890045, x1 = 836564809606, x2 = 942785024683 is the current record in terms of the number of digits.