N-bonacci Sequences - Taking Fibonacci Further
Leonardo Fibonacci (c.1170 - c.1250)
Fibonacci Numbers
The Fibonacci sequence is one of the best known sequences of numbers in mathematics and was first brought to attention in Europe by Leonardo of Pisa (often known by the name 'Fibonacci') in his 1202 work Liber Abaci.
He was tackling a problem about rabbit populations with immortal rabbits who took one month after birth to mature, and then continued to produce one pair of young rabbits each month. Each new pair of rabbits also took one month to mature before then also producing a new pair of young rabbits each month.
He discovered that the number of pairs of rabbits each month followed a very interesting sequence which we call the Fibonacci sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …
In short, this sequence is derived by starting with two ones and then finding each term by adding the previous two terms together. For example, 1+1=2, 1+2=3, 2+3=5 and so on.
This sequence has a lot of interesting properties including its link to the Golden ratio, but the thing we are going to look at in this article is what happens if, instead of adding two terms together to get the next, you add a larger number of terms together.
He was tackling a problem about rabbit populations with immortal rabbits who took one month after birth to mature, and then continued to produce one pair of young rabbits each month. Each new pair of rabbits also took one month to mature before then also producing a new pair of young rabbits each month.
He discovered that the number of pairs of rabbits each month followed a very interesting sequence which we call the Fibonacci sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …
In short, this sequence is derived by starting with two ones and then finding each term by adding the previous two terms together. For example, 1+1=2, 1+2=3, 2+3=5 and so on.
This sequence has a lot of interesting properties including its link to the Golden ratio, but the thing we are going to look at in this article is what happens if, instead of adding two terms together to get the next, you add a larger number of terms together.
The 3-Bonacci Sequence
The 3-Bonacci sequence (or Tribonacci sequence) works by putting an extra zero at the beginning before the two 1s and this time finding the next term by adding the three previous terms together. This gives us the following sequence:
0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149 …
0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149 …
The 4-Bonacci Sequence
The 4-Bonacci sequence (or Tetranacci sequence) is defined by placing two zeros at the beginning and this time adding four terms together to get the next one. The first few terms of the 4-Bonacci sequence are:
0, 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, …
0, 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, …
Higher Orders of Fibonacci's Sequence (N-Bonacci Sequences)
Our list of sequences can be extended further by adding different numbers of terms together. People have calculated Pentanacci numbers by adding five terms together, Hexanacci by using six terms and many more. In fact you can create a Fibonacci variant using any positive whole number.
We call these sequences the N-bonacci numbers, where N stands for the number of terms added together in order to create the next term. I have listed the names of the first thirteen in the table below.
We call these sequences the N-bonacci numbers, where N stands for the number of terms added together in order to create the next term. I have listed the names of the first thirteen in the table below.
N |
Name of sequence |
Medenacci |
|
1 |
Enanacci |
2 |
Fibonacci |
3 |
Tribonacci |
4 |
Tetranacci |
5 |
Pentanacci |
6 |
Hexanacci |
7 |
Heptanacci |
8 |
Octanacci |
9 |
Enneanacci |
10 |
Decanacci |
11 |
Hendecanacci |
12 |
Dodecanacci |
Medenacci and Enanacci Sequences
Eagle-eyed readers will have spotted two sequences at the top of the table before the Fibonacci sequence. These are the Medenacci sequence where you add together zero previous terms and the Enanacci sequence where you add together one previous term.
These are degenerate cases of the N-bonacci sequence (degenerate cases are a limiting case where the properties appear to be different to the rest of the group).
The Medenacci sequence is:
0, 0, 0, 0, 0, 0, 0, 0, …
and the Enanacci sequence is:
1, 1, 1, 1, 1, 1, 1, 1, …
These are degenerate cases of the N-bonacci sequence (degenerate cases are a limiting case where the properties appear to be different to the rest of the group).
The Medenacci sequence is:
0, 0, 0, 0, 0, 0, 0, 0, …
and the Enanacci sequence is:
1, 1, 1, 1, 1, 1, 1, 1, …
Just For Fun
See if you can find the first few terms of the Pentanacci, Hexanacci or any other N-bonacci sequence. As a starting point, make sure you begin your sequence with N − 2 zeroes i.e. the Hexanacci will need to begin with 6 − 2 = 4 zeroes before the two 1s.
Comments
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