The Fibonacci sequence is named after Leonardo Pisano Fibonacci.

Most of us have heard of the Fibonacci sequence. You start with the numbers 0 and 1 and generate subsequent terms by taking the sum of the two previous ones, giving you the infinite sequence

$0,1,1,2,3,5,8,13,21,34,55,89,144...$The 3-bonacci sequence is a variation on this. Start with the numbers 0, 0, and 1, and generate subsequent numbers by taking the sum of the three previous terms. This gives the infinite sequence

$0,0,1,1,2,4,7,13,24,44,81,149,274,... $The 4-bonacci sequence starts with the numbers 0,0,0, and 1. Every subsequent number is generated by taking the sum of the four previous terms:

$0, 0, 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, 208,...$ You can see how this can be generalised further: the $N$-bonacci sequence starts with $N-1$ zeroes followed by a 1, with subsequent terms being generated by taking the sum of the $N$ previous terms.The traditional Fibonacci sequence corresponds to the 2-bonacci sequence. And the 1-bonacci sequence consists entirely of 1s.

*N*-bonacci constants

The Fibonacci sequence has a famous property. If you divide each term by the term that comes before it, you get a sequence of ratios:

$$1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, 55/34, 89/55, 144/89,...$$(This is ignoring the ratio between the first two terms because division by 0 isn't defined.) This sequence of ratios converges to the famous golden ratio,

$$\phi=\frac{1+\sqrt{5}}{2}=1.61803398875...$$ You can find out more about this here. Is something similar true for other $N$-bonacci sequences? The answer is "yes", and the corresponding limits are called*$N$-bonacci constants*(or sometimes $N$-

*anacci constants*). Here are first five $N$-bonacci constants:

N | N-bonacci constant |

1 | 1 |

2 | 1.61803... |

3 | 1.83928... |

4 | 1.92756... |

5 | 1.96595... |

### The infinacci sequence

We can even talk about the *infinacci sequence*, for which $N$ equals infinity. This starts with an infinite number of 0s, followed by a 1. The next term is the sum of the infinitely many initial 0s (which is 0) and 1, giving us a 1. Subsequent terms are

*infinacci constant*, is also equal to $2.$ This fits with equation (1) above. Suppose you let the $N$ in this equation tend to infinity. Then, as long as $x > 1$ (which we can assume as we are considering solutions greater than $1$), the term $1/x^N$ in the equation tends to 0. This means that the solution to the equation that is greater than $1$ tends to $x=2$.

Finally, here's another interesting bit of information. If you know about the Fibonacci sequence then you probably know that Fibonacci came up with it while thinking of a hypothetical population of rabbits (find out more here). It has been suggested that the tribonacci sequence also has an animal connection. In *On the origin of species* Charles Darwin considers the growth of a population of animals based on a calculation made by his son George H. Darwin — and it has been suggested that this calculation involves the tribonacci sequence.