TITLE: Interesting properties of Fibonacci-like sequences?
QUESTION [13 upvotes]: Everyone is familiar with the Fibonacci Sequence, [0] 1 1 2 3 5 8 ... and many of it's interesting properties. For example, as the sequence continues, the ratio of $\frac{F_n}{F_{n-1}}$ converges to $\tau=\frac{1+\sqrt{5}}{2}$, a ratio which can be used to describe a number of numerical relationships in nature. 
From some quick Wikipedia browsing, we can find a number of Generalizations of Fibonacci numbers. For one abstract generalization, we can define Fibonacci-like sequences as follows:
An $n$-order Fibonacci-like sequence is generated by $F_n=\sum_{i=1}^{n}F_{n-i}$ with $n$ initial terms. 
Thus, the Fibonacci sequence is such a sequence with $n=2$ and $F_0=0$ and $F_1=1$
Using this basic generalization, we have Lucas Numbers, where $n=2$ and $F_0=2$ and $F_1=1$, whose consecutive-number ratio also converges to the golden ratio. There are also Tribonacci, Tetranacci, and n-nacci numbers, which follow this generalization for 3, 4, and n numbers, and pad the initializing values with 0s, e.g. $F_0=0$ ... $F_{n-2}=0$, $F_{n-1}=1$.
So, my question is, are there any important properties of these sequences that are worth learning? Do these sequences have real-world applications or reflections like the original Fibonacci sequence does? What can be learned from these?

REPLY [4 votes]: I can think of at least eight interesting things about the tribonacci numbers and its limiting ratio, the tribonacci constant. Whereas for the Fibonacci numbers the discriminant $\sqrt{5}$ plays a role, for the tribonacci it is $\sqrt{11}$. 
I. Sequences 
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. 
II. Powers 
Let $a=\tfrac{1}{3}(19+3\sqrt{33})^{1/3},\; b=\tfrac{1}{3}(19-3\sqrt{33})^{1/3}$, then powers of the tribonacci constant $T$ can be expressed in terms of those three sequences as,
$$3T^n  = R_{n}+(a+b)S_{n-1}+3(a^2+b^2)T_{n-1}$$ 
III. q-Continued fraction 
Let $q = -1/(e^{\pi\sqrt{11}})$, then,
$$\frac{(e^{\pi\sqrt{11}})^{1/24}}{\frac{1}{T}+1} = 1 + \cfrac{q}{1-q + \cfrac{q^3-q^2}{1 + \cfrac{q^5-q^3}{1 + \cfrac{q^7-q^4}{1 + \ddots }}}}$$
IV. Snub cube
The Cartesian coordinates for the vertices of a snub cube are all the even permutations of $v=1/T$,
$\hskip2.8in$ 
V. Pi Formula
Let $v = 1/T$, then,
$$\small\sum_{n=0}^\infty \frac{(2n)!^3}{n!^6}\frac{2(4v+1)(2v+1)n+(4v^2+2v-1)}{(v+1)^{24n}} =\frac{4}{\pi}$$
VI. Infinite Nested Radical
$$\frac{1}{T-1} = \sqrt[3]{\frac{1}{2}+ \sqrt[3]{\frac{1}{2}+ \sqrt[3]{\frac{1}{2}+ \sqrt[3]{\frac{1}{2}+\dots}}}}$$
VII. Complete elliptic integral of the first kind $K(k_{11})$
Alternatively, its exact value is,
$$K(k_{11}) = \frac{1}{11^{1/4}(4\pi)^2} \bigl(\tfrac{T+1}{T}\bigr)^2\; \Gamma\bigl(\tfrac{1}{11}\bigr) \Gamma\bigl(\tfrac{3}{11}\bigr) \Gamma\bigl(\tfrac{4}{11}\bigr) \Gamma\bigl(\tfrac{5}{11}\bigr) \Gamma\bigl(\tfrac{9}{11}\bigr) = 1.570983\dots$$
VIII. Cubic Pell-Type Equation
The diophantine cubic Pell-type equation, 
$$a^3 - 2 a^2 b + 2 b^3 - a^2 c - 2 a b c + 2 b^2 c + a c^2 + 2 b c^2 + c^3=1$$
has an infinite number of integer solutions, 
$$a,\;b,\;c = T_{n-1},\;T_{n-2},\;T_{n-3}$$
Conclusion: Surely the tribonacci numbers are interesting enough?
P.S. See also this post.