Starting with u^(0)=0, u^(1)=0, u^(2)=1, define the sequence...

Starting with $u^{(0)}=0, u^{(1)}=0, u^{(2)}=1$, define the sequence of tribonacci numbers $u^{(k)}$ by adding the previous three to get the next one. For instance, $u^{(3)}=u^{(0)}+u^{(1)}+u^{(2)}=1$. (a) Write out the next four tribonacci numbers. (b) Find a third order iterative equation for the $k^{\text {th }}$ tribonacci number. (c) Explain why the tribonacci numbers are all integers. (d) Find an explicit formula for the solution, using a computer to approximate the eigenvalues. (c) Do they grow faster than the usual Fibonacci numbers? What is their overall rate of growth?

Key Concepts

Tribonacci Sequence

The Tribonacci sequence is a generalization of the Fibonacci sequence in which each term is the sum of the three preceding terms. It is defined recursively with initial values specified and illustrates the idea of building a sequence from a fixed number of previous outputs. This sequence is useful for understanding higher-order recurrences and the behavior of such recurrences under different starting conditions.

Recurrence Relation

A recurrence relation expresses each term of a sequence as a function of its predecessors. For the Tribonacci sequence, the recurrence relation is a third order linear homogeneous recurrence relation with constant coefficients. This means that each term can be computed as a linear combination of the previous three terms, establishing a systematic method to build the sequence iteratively.

Third Order Iterative Equation

A third order iterative equation is an equation that defines a sequence where each term depends on the three preceding terms. This iterative nature means that once the initial three terms are given, every subsequent term is determined by repeatedly applying the same recurrence rule. This is important for developing iterative algorithms and understanding the dynamics of systems modeled by higher-order linear recurrences.

Characteristic Equation and Eigenvalue Analysis

In solving linear recurrence relations with constant coefficients, one common method is to assume a solution of the form r^k and derive the characteristic equation. For a third order recurrence like the Tribonacci sequence, this yields a cubic equation whose roots, or eigenvalues, determine the general form of the sequence. These eigenvalues, once approximated, provide crucial information for formulating an explicit closed-form solution and analyzing the long-term behavior of the sequence.

Proof of Integrality

The observation that the Tribonacci numbers are all integers can be justified by the principle of mathematical induction. Since the initial conditions are integers and the recurrence relation involves summing three integers to produce the next term, it follows that every term in the sequence will be an integer. This illustrates a broader concept in number theory concerning sequences defined by integer recurrences.

Exponential Growth Rate

The growth rate of sequences defined by linear recurrences is often governed by the largest absolute value among the eigenvalues from the characteristic equation. For the Tribonacci sequence, it has been shown that the terms grow exponentially with a rate determined by this dominant eigenvalue. This means that as the sequence progresses, its values increase at a rate faster than many other sequences like the traditional Fibonacci numbers, highlighting an important concept in the asymptotic analysis of recursive sequences.

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