Triple Fibonacci Sequence

Using the recursive relation for a sequence similar to the Fibonacci sequence, where $a_0 = 1$, $a_1 = 1$, $a_2 = 1$, and $a_n = a_{n-1} + a_{n-2} + a_{n-3}$, find the first few terms of the sequence.

This problem deals with a variant of the classic Fibonacci sequence by introducing a recursive relationship that sums three previous terms instead of the usual two. This variant is particularly interesting because it showcases how small modifications in a recursive definition can significantly alter the nature and behavior of a sequence. Understanding this problem requires a grasp of recursive definitions and how sequences can be extended from initial values using these definitions.

In this sequence, you start with three initial conditions, where each are set to one: a sub 0, a sub 1, and a sub 2. The recursive formula a sub n equals a sub n minus 1 plus a sub n minus 2 plus a sub n minus 3, introduces the challenge of calculating terms from a dynamic base, rather than a static base of two terms as seen in the Fibonacci sequence.

This problem provides practice with recursive relations, highlighting the strategy of using previously calculated terms to establish new terms. Solving this problem will reinforce the understanding of recursive formulas, which are critical in computer science for algorithm design, especially in dynamic programming approaches. In learning how to extend a given sequence, one appreciates the power and flexibility of recursion in mathematical contexts.