$ 1-\frac{1}{x} $, the Unit Circle and Reciprocal of the Tribonacci Constant

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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

$$ \begin{align} x_1 &= 1 \\[5pt] x_2 &= \frac{1}{3} \left( -1 - \frac{2}{\sqrt[3]{17+3\sqrt{33}}} + \sqrt[3]{17+3\sqrt{33}} \right) = 0.54368901 \ldots \\[5pt] x_3 &= \frac{1}{6} \left( -2 + \frac{2-2i\sqrt{3}}{\sqrt[3]{17+3\sqrt{33}}} + (-1-i\sqrt{3}) \sqrt[3]{17+3\sqrt{33}} \right) \\[5pt] x_4 &= \frac{1}{6} \left( -2 + \frac{2+2i\sqrt{3}}{\sqrt[3]{17+3\sqrt{33}}} + i(\sqrt{3} + i) \sqrt[3]{17+3\sqrt{33}} \right) \text. \end{align} $$

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.