Characteristic polynomials for $K$-Bonacci numbers: what's their name? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T06:00:04Z http://mathoverflow.net/feeds/question/47140 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/47140/characteristic-polynomials-for-k-bonacci-numbers-whats-their-name Characteristic polynomials for $K$-Bonacci numbers: what's their name? ccarminat 2010-11-23T19:17:57Z 2010-11-23T23:10:41Z <p>Fibonacci numbers are defined by the recurrence relation $f_{n+2}=f_{n+1}+f_{n}$ and Tribonacci numbers by $f_{n+3}=f_{n+2}+f_{n+1}+f_{n}$</p> <p>One can define, in general, K-Bonacci numbers as $f_{n+K}=f_{n+K-1}+...+f_{n+1}+f_{n}$</p> <p>(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).</p> <p>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.</p> <p>Question: <strong>do these polynomials have a already a name?</strong></p> http://mathoverflow.net/questions/47140/characteristic-polynomials-for-k-bonacci-numbers-whats-their-name/47144#47144 Answer by Gottfried Helms for Characteristic polynomials for $K$-Bonacci numbers: what's their name? Gottfried Helms 2010-11-23T20:15:18Z 2010-11-23T20:20:33Z <p>A very good reference for this is at <a href="http://www.dur.ac.uk/bob.johnson/fibonacci/" rel="nofollow">Bob Johnson's fibonacci-page</a> the article <a href="http://www.maths.dur.ac.uk/~dma0rcj/PED/fib.pdf" rel="nofollow">Fibonacci and matrices</a> Bob deals with the generalization in terms of matrix-representation.  Well, second thought: I realize: don't know whether he introduces a name for it...</p> http://mathoverflow.net/questions/47140/characteristic-polynomials-for-k-bonacci-numbers-whats-their-name/47154#47154 Answer by Nikita Sidorov for Characteristic polynomials for $K$-Bonacci numbers: what's their name? Nikita Sidorov 2010-11-23T22:13:20Z 2010-11-23T22:29:57Z <p>The dominant root of such a polynomial is often referred to as a <strong>multinacci number</strong>. These numbers are known to be <a href="http://mathworld.wolfram.com/PisotNumber.html" rel="nofollow">Pisot numbers</a> and, indeed, tend to 2. </p> http://mathoverflow.net/questions/47140/characteristic-polynomials-for-k-bonacci-numbers-whats-their-name/47159#47159 Answer by Gerry Myerson for Characteristic polynomials for $K$-Bonacci numbers: what's their name? Gerry Myerson 2010-11-23T23:10:41Z 2010-11-23T23:10:41Z <p>You could call them the A154990-polynomials, <a href="http://oeis.org/A154990" rel="nofollow">http://oeis.org/A154990</a></p>