In this 3-part series of posts, we’ll discuss well-known sequences with the recurrence,

where {*a, b, c*} *can only be zero or unity*. Aside from the *Fibonacci* and *Lucas numbers* which is *a* = 0, there is the *Narayana sequence* with *b* = 0, the *Padovan *and* Perrin *with *c* = 0, and the *tribonacci* has *a = b = c* = 1. All four cases may then share similar properties and one of which, interestingly enough, is that their *limiting ratios*, a root of the following equations,

can also be used to express , or *Apery’s constant*.

**I. Fibonacci and Lucas numbers**

Given the two roots of,

with , the larger root being the *golden ratio*, we get the *Lucas numbers* *L*(n) and *Fibonacci numbers* *F*(n),

(The starting index is *n* = 0.) Expanding powers of the golden ratio, then for *n* > 0,

We’ll see this can be generalized to powers of the *tribonacci constant*.

**II. Tribonacci numbers **

These are a generalization of the Fibonacci numbers, being,

Pin-Yen Lin has a nice paper involving these numbers. First, define the following three sequences with this recurrence, but with different initial values,

(The starting index as usual is *n* = 0.) The first and the third are recognized by the OEIS, with the first being *the* tribonacci numbers. The limiting ratio for all three is the *tribonacci constant*, *T*, the real root of,

or,

I’ve already written about the tribonacci constant before. But I want to include how Lin found that powers of *x* can be expressed in terms of those three sequences. Define,

then, *similar to the golden ratio*,

Hence, starting with *n *= 1,

and so on. Interesting, isn’t it, that powers of the tribonacci constant can be expressed in this manner.

*Addendum*:

There is a primality test regarding Lucas numbers: if *n* is a prime then is divisible by *n*. For example , minus 1, is divisible by 5. However there are *Lucas pseudoprimes*, composite numbers that pass this test, with the smallest being *n* = 705.

The third tribonacci sequence can be formed analogously to the Lucas numbers. Given the three roots of,

then, starting with *n* = 0,

I notice that likewise, if *n* is prime, then is divisible by *n*. But there are also *tribonacci-like pseudoprimes*. The smallest is *n* = 182. Steven Stadnicki was nice enough to compute the first 36. It turns out they are relatively rarer, as there are only 21 less than , while there are 852 Lucas pseudoprimes in the same range.