**Abstract**: We show that the equation \( y = 1-\frac{1}{x} \), the unit circle and reciprocal of the tribonacci constant are all related at an intersction between \( y = 1-\frac{1}{x} \) and the unit circle.

#### What are the tribonacci numbers?

The tribonacci numbers are defined by the recurrence relation

$$ q_n = q_{n-1} + q_{n-2} + q_{n-3} $$where the initial terms are \( q_0 = 0 \), \(q_1=0\) and \( q_2=1 \) [1]. The first few tribonacci numbers are:

$$ 0,0,1,1,2,4,7,13,24,44,81,149,274, 504, 927, 1705, 3136, 5768, \ldots $$(sequence A000073 in the OEIS)

The name tribonacci was given by a brilliant fourteen-year old junior high student by the name of Mark Feinberg in his paper "Fibonacci-Tribonacci" published by *Fibonacci Quarterly* in 1963 [1].

"The series was first described formally by Agronomof in 1914" [Wikipedia] but an earlier documented case of its use can be traced to Charles Darwin in the *Origin of Species* to describe the hypothetical population growth of elephants [2]. The problem limited reproductive elephants to three consecutive generations.

Like the Fibonacci numbers, the tribonacci numbers are convergent. The ratio of consecutive terms \( \frac{q_{n+1}}{q_n} \) converges to \( 1.83928675 \ldots \) which is known as the tribonacci constant [1]. The reciprocal of the tribonacci constant is \( 0.54368901 \ldots \).

#### Proof

We first define the two equations that are to intersect. The first equation is

$$ y = 1 - \frac{1}{x} \tag{1} $$and the second equation is for the unit circle

$$ x^2 + y^2 = 1 \text. \tag{2} $$To find the intersections between \( y = 1-\frac{1}{x} \) and the unit circle we substitute equation (1) into equation (2),

$$ x^2 + \left(1-\frac{1}{x}\right)^2 = 1 $$and the equation we find is \( x^4 - 2x + 1 = 0 \text. \)

Solving for \(x\) and using WolframAlpha we have

The intersection points are \( (1, 0) \) and \( (0.543689, -0.839286) \).

To check our answer we substitute \( x = 0.543689 \) and \( y = -0.839286 \) in (2) and we have

$$ 0.543689^2 + 0.839286^2 = 1 \text. $$#### Conclusion

The indication of a potential pattern emerges since the golden mean conjugate (reciprocal) is found at the intersection of \(x^2\) and the unit circle. There are likely equations that intersect with the unit circle at coordinate(s) related to the reciprocal constants of tetranacci and higher-order Fibonacci sequences.

#### References

[1] Mark Feinberg, Fibonacci-Tetranacci, *The Fibonacci Quarterly,* **1.3**, 71-74 (1963).

[2] Podani, János; Kun, Ádám; Szilágyi, András, "How Fast Does Darwin's Elephant Population Grow?" (PDF). *Journal of the History of Biology,* **51.2**, 259–281 (2018). doi:10.1007/s10739-017-9488-5

#### Note

As some might be curious as to where Mark Feinberg is now, tragically four years after publishing his paper he passed away due to a motorcycle accident at the age of eighteen.