Derived sequences, the Tribonacci recurrence and cubic forms. (English) Zbl 1001.11008

Elia, Michele

Fibonacci Q. 39, No. 2, 107-115 (2001).

Integer representations by forms are sources of very interesting Diophantine equations and are discussed here by a cubic form associated with a third-order recurrence known as the Tribonacci recurrence; i.e., \(T_{n+3} = T_{n+2} + T_{n+1} + T_n\). The main properties of the Tribonacci recurrence and the Tribonacci ring are studied. Then the integer representation problem for the Tribonacci cubic form is fully solved.

Reviewer: Edward L. Cohen (Ottawa)

MSC:

11B39	Fibonacci and Lucas numbers and polynomials and generalizations
13G05	Integral domains
39A99	Difference equations

Keywords:

integer representations; Tribonacci recurrences; cubic forms