A number $N$ is a $k$-nacci number if and only if ...

Note: Sequences start with $n=0$.

I. $k = 2$

The Lucas numbers,

$$L_n = x_1^n+x_2^n = 2,1,3,4,7,11,18,29,\dots$$

and the Fibonacci numbers,

$$F_n = \frac{x_1^n}{y_1}+\frac{x_2^n}{y_2} = 0,1, 1, 2, 3, 5, 8, 13, 21,\dots$$

where $y_i =2x_i-1$ and the $x_i$ are the roots of $x^2-x-1=0$. Then,

$$x_1^n = \big(\tfrac{1+\sqrt{5}}{2}\big)^n=\tfrac{1}{2}(L_n+F_n\sqrt{5})$$

$$x_2^n = \big(\tfrac{1-\sqrt{5}}{2}\big)^n=\tfrac{1}{2}(L_n-F_n\sqrt{5})$$

Hence,

$$(x_1 x_2)^n = (-1)^n = \tfrac{1}{4}(L_n^2-5F_n^2)$$

II. $k = 3$

(Edited.) Given three sequences with recurrence $s_n = s_{n-1}+s_{n-2}+s_{n-3}$ but different initial values as,

$$\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \text{Name} & \text{Formula} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & OEIS\\ \hline R_n & x_1^n+x_2^n+x_3^n &3 &1 &3 &7 &11 &21 &39 & 71 & A001644 \\ \hline S_n &\frac{x_1^n}{y_1}+\frac{x_2^n}{y_2}+\frac{x_3^n}{y_3}&3 &2 &5 &10 &17 &32 &59 &108 &(none)\\ \hline T_n &\frac{x_1^n}{z_1}+\frac{x_2^n}{z_2}+\frac{x_3^n}{z_3}&0 &1 &1 &2 &4 &7 &13 &24& A000073 \\ \hline \end{array}$$

$$y_i =\tfrac{1}{19}(x_i^2-7x_i+22)$$ $$z_i =-x_i^2+4x_i-1$$

and the $x_i$ are the roots of $x^3-x^2-x-1=0$, with $T_n$ as the tribonacci numbers and the real root $T = x_1 \approx 1.83929$ the tribonacci constant. Define,

$$a=\tfrac{1}{3}(19+3\sqrt{33})^{1/3},\quad b=\tfrac{1}{3}(19-3\sqrt{33})^{1/3}$$

then we have the nice identity by Pin-Yen Lin,

$$3T^n = R_{n}+(a+b)S_{n-1}+3(a^2+b^2)T_{n-1}$$

and similar expressions for the complex conjugates $x_2^n$ and $x_3^n$ (after correcting some typos in the paper). Hence it is possible to express,

$$(-x_1x_2x_3)^n = (1)^n = \text{in terms of}\; R_{n},\, S_{n-1},\, T_{n-1}$$

analogous to the $k=2$ case, and get rid of irrationalities. For some reason, Lin didn't bring it to this step so we find it with the help of Mathematica. Since $R_n,\;S_n$ can be expressed as sums of $T_m$, define,

$$a,\;b,\;c = T_{n-1},\;T_{n-2},\;T_{n-3}$$

then we get the diophantine cubic Pell-type equation for the tribonacci numbers,

$$a^3 - 2 a^2 b + 2 b^3 - a^2 c - 2 a b c + 2 b^2 c + a c^2 + 2 b c^2 + c^3=1$$

with an infinite number of integer solutions analogous to the Pell equation for the Fibonacci numbers,

$$p^2-5q^2 = \pm 4.$$

III. $k = 4$

Define sequence A073817,

$$U_n = x_1^n+x_2^n+x_3^n+x_4^n = 4, 1, 3, 7, 15, 26, 51, 99, 191,\dots$$

and the tetranacci numbers,

$$V_n = \frac{x_1^n}{y_1}+\frac{x_2^n}{y_2}+\frac{x_3^n}{y_3}+\frac{x_4^n}{y_4} =0,1, 1, 2, 4, 7, 13, 24, 44, 81,149\dots$$

where $y_i =-x_i^3 + 6x_i - 1$ and the $x_i$ are the roots of $x^4-x^3-x^2-x-1=0$. Then,

$$\text{(insert relationship here)}$$

A number $N$ is Tribonacci if and only if it can be written as:

$$N= \frac{\alpha^n}{(\alpha-\beta)(\alpha-\gamma)}+\frac{\beta^n}{(\beta-\alpha)(\beta-\gamma)}+\frac{\gamma^n}{(\gamma-\alpha)(\gamma-\beta)}$$

Where $n\geq 1$ and $\alpha, \beta,\gamma$ are the three roots of the polynomial $$x^3-x^2-x-1$$

Source: T. Noe, T. Piezas and E. Weisstein, Tribonacci Number (see here).