Characteristic polynomials for $K$-Bonacci numbers: what's their name? - MathOverflow

Characteristic polynomials for $K$-Bonacci numbers: what's their name?

2010-11-23T19:17:57Z

Fibonacci numbers are defined by the recurrence relation $f_{n+2}=f_{n+1}+f_{n}$ and Tribonacci numbers by $f_{n+3}=f_{n+2}+f_{n+1}+f_{n}$

One can define, in general, K-Bonacci numbers as $f_{n+K}=f_{n+K-1}+...+f_{n+1}+f_{n}$

(they show up naturally if you consider the problem of counting binary strings of length n which do not contain sequences of K adjacent zeroes).

The characteristic polynomial associated to K-Bonacci numbers is $$P_K(t):=t^K-(t^{K-1}+t^{K-2}+...+t+1)$$ By the way, he same polynomial turns up when trying to calculate the asymptotic growth rate via generating functions and, as $K \to +\infty$, the biggest real root approaches 2.

Question: do these polynomials have a already a name?

Answer by Gottfried Helms for Characteristic polynomials for $K$-Bonacci numbers: what's their name?

2010-11-23T20:15:18Z

A very good reference for this is at Bob Johnson's fibonacci-page the article Fibonacci and matrices Bob deals with the generalization in terms of matrix-representation. [edit] Well, second thought: I realize: don't know whether he introduces a name for it...

Answer by Nikita Sidorov for Characteristic polynomials for $K$-Bonacci numbers: what's their name?

2010-11-23T22:13:20Z

The dominant root of such a polynomial is often referred to as a multinacci number. These numbers are known to be Pisot numbers and, indeed, tend to 2.

Answer by Gerry Myerson for Characteristic polynomials for $K$-Bonacci numbers: what's their name?

2010-11-23T23:10:41Z

You could call them the A154990-polynomials, http://oeis.org/A154990