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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?

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    $\begingroup$ If you need to refer to these polynomials, you could call them the "Characteristic polynomials of the K-Bonacci numbers". ;-) $\endgroup$
    – André Henriques
    Commented Nov 23, 2010 at 19:48

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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.

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  • $\begingroup$ Your link does not work for me. Looking at your source, it seems you did it the right way, so I'm not sure what could be wrong. $\endgroup$
    – Will Jagy
    Commented Nov 23, 2010 at 22:25
  • $\begingroup$ @Will, I changed it from Wiki to Mathworld. Basically, a Pisot number is an algebraic integer >1 whose other Galois conjugates are less than 1 in modulus. Like the golden ratio, for instance. These numbers play an important role in dynamics, measure theory and tilings. $\endgroup$
    – Nikita Sidorov
    Commented Nov 23, 2010 at 22:32
  • $\begingroup$ That's better. There is a way to get the most convoluted Wikipedia address working as a link, but I'm not very good at it. I also was hoping to work in a comment about multinational conglomerates, but sometimes the words just do not come together. My policy remains that if the joke needs either an explanation or an emoticon it is not worth writing. Irving Kaplansky and I did some work on these, never published. If I can find a good printout I will scan it in. That does not guarantee there is anything unknown in the notes. $\endgroup$
    – Will Jagy
    Commented Nov 23, 2010 at 23:07
  • $\begingroup$ Thanks, Nikita. You could well suggest to have a look at your pages, which seem to be full of interesting stuff (which goes far beyond this question). Strictly speaking the problem of "finding the name" is still unanswered; neverthanless the name was just a mean to find connections to other topics. Bye $\endgroup$
    – ccarminat
    Commented Nov 23, 2010 at 23:29
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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...

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

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