RootSum Sequences

[Mitch Harris]

From: Ed Pegg Jr [mailto:edp at wolfram.com]
Sent: Thursday, May 25, 2006 1:13 PM
> 
> Start with a polynomial with integer coefficients, and find 
> it's roots.  
> The sum of these roots, and the
> sums of the powers of these roots, is an integer.
> 
> So, some examples, starting with x^3 - 6 x^2 + 4 x - 2.
> 
> Table[RootSum[#^3 - 6 #^2 + 4 # - 2 &,#^n &],{n,1,20}]
> 
> 6, 28, 150, 800, 4256, 22636, 120392, 640320, 3405624, 18113248, 

...
> Are there any polynomials that give some OEIS Sequences?  

Of course! You just described the generalization of the Lucas numbers
(the relative of the Fibonacci numbers with different base cases)

A polynomial with integer coefficients is the characteristic equation of
a linear recurrence relation with integer coefficients, the latter of
which necessarily has an integer sequence solution.

Given the roots r_1, r_2,... of the polynomial, the Lucas sequence (and
its generalizations) has r_1^n + r_2^n + ... as its closed form
solution.

Table[RootSum[#^3 - #^2 - # - 1 &, #^n &], {n, 0, 20}]

gives in the OEIS:

http://www.research.att.com/~njas/sequences/A001644

for the recurrence a[n] = a[n-1] + a[n-2] + a[n-3] (the base cases are
forced by the root sum's first three values).


Mitch