What is the Fibonacci-like sequence called where one sums the last 3 numbers

Question

The Fibonacci-sequence is defined like.

$F_{x+1} = F_{x} + F_{x-1}; F_0 = 0, F_1=1, x \in {\Bbb N}$

Is there a special name for this sequence:

$F_{x+1} = F_{x} + F_{x-1} + F_{x-2}$ ?

Which?

https://en.wikipedia.org/wiki/Generalizations_of_Fibonacci_numbers#Tribonacci_numbers — Jack D'Aurizio, May 24 '16 at 11:39
As discussed here, this is often called the tribonacci sequence. — hardmath, May 24 '16 at 11:47
Both of you are right. Thanks. as @JackD'Aurizio was the first... Please make it an answer. — steffen, May 24 '16 at 18:21

score 2 · Accepted Answer · answered May 24 '16 at 18:34

2

The sequence given by $T_0=1,T_1=T_2=1$ and $T_{n+3}=T_{n+2}+T_{n+1}+T_n$ is usually called the tribonacci sequence. It shares many properties with the Fibonacci sequence (like the existence of a Zeckendorf-like theorem) but obviously has a different characteristic polynomial, $x^3-x^2-x-1$.

answered May 24 '16 at 18:34

Jack D'Aurizio

353,855

According to your link T_2=2 not 1. Which makes sense because if it was 1 then T_2 != T_0+T_1 – A Kareem Dec 14 '21 at 07:52
Well, a shift of index does not affect the characteristic polynomial. – Jack D'Aurizio Dec 14 '21 at 11:32

score 1 · Answer 2 · answered Aug 01 '16 at 23:40

There are higher fibonacci sequences. The one you mentioned is called a Tribonacci. Higher ones are Tetranacci, Pentanacci, Hexanacci, Heptanacci, etc. You can either recursively generate the sequence or use a characteristic polynomial for each. More info here (at the bottom of page): http://www.genautica.com/math/naccis/fibonacci_and_higher_order_naccis.html

To get the first 100 numbers of your Tribonacci with Python, you could do something like this:

list1=[1,2,3]

for _ in xrange(4,101):
    list1.append(list1[-3]+list1[-2]+list1[-1])

print list1

What is the Fibonacci-like sequence called where one sums the last 3 numbers

2 Answers2