Abstract
In this note, we derive Binet's formula for the general term of the generalized tribonacci sequence. This formula gives
explicitly as a function of the index n, the roots of the associated characteristic equation, and the initial terms
,
, and
. By way of illustration, we obtain Binet's formula for the Cordonnier, Perrin, and Van der Laan numbers. In addition, we establish a double identity that can be regarded as a parent of Binet's formula for generalized tribonacci numbers.
Acknowledgements
The author thanks an anonymous reviewer for detailed instructive suggestions that led to improvements in the presentation of this note.
Disclosure statement
No potential conflict of interest was reported by the author.
Notes
1. Unfortunately, Mark Feinberg died in a motorcycle accident four years later.
2. The closed-form formula for the nth Fibonacci number, (where
and
), is usually attributed to the French mathematician Jacques Philippe Marie Binet (1786–1856) who published it in 1843, although it was known in the eighteenth century, to such famous mathematicians as Abraham de Moivre (1667–1754), Daniel Bernoulli (1700–1782), and Leonhard Euler (1707–1783). Amazingly enough, the exponential growth of the Fibonacci numbers played a prominent role in the resolution of Hilbert's 10th problem in 1970 by the combined efforts of Yuri Matiyasevich, Martin Davis, Hilary Putnam, and Julia Robinson.
3. Throughout this note, we will assume that α ≠ β, α ≠ γ, and β ≠ γ. This is tantamount to assuming that Δ, the discriminant, is not zero. That is, Δ2 = (α − β)2(α − γ)2(β − γ)2 = r2s2 − 27t2 + 4s3 − 4r3t − 18rst > 0.
4. Cordonnier numbers are more commonly known in the literature as Padovan numbers.
6. From Equations (Equation10a(10a)
(10a) ) and (Equation10b
(10b)
(10b) ), it is not difficult to show that Pn(α, β, γ) and Qn(α, β, γ) are symmetric functions of α, β, and γ. We omit the proof here for the sake of brevity (although the reader might want to prove that, for example, Pn(α, β, γ) = Pn(β, α, γ)). Moreover, from the expressions (Equation10a
(10a)
(10a) ) and (Equation10b
(10b)
(10b) ), it is readily seen that Pn(β, α, γ) = Qn(α, β, γ), from which we conclude that Pn(α, β, γ) = Qn(α, β, γ). Specifically, for n = 1, 2, and 3, we have
which are clearly symmetric functions of α, β, and γ.