RootSum Sequences

[Antti Karttunen]

franktaw at netscape.net wrote:

> Well, there's A000032, the Lucas numbers, for x^2 - x - 1; A002203 for 
> x^2 - 2x - 1; except for the initial term, A084099 for x^2 - 1; also 
> except for the initial term, A099837 for x^2 + x + 1; A080040 for x^2 
> - 2x - 2; dropping two initial terms, A077957 for x^2 - 2; A001644 for 
> x^3 - x^2 - x - 1.
>
> Always try the simple cases first.
>
> Franklin T. Adams-Watters
>

Here's a project:

Collect all fractions from Stern-Brocot tree:

A007305: 1,1,2,1,2,3,3,1,2,3,3,4,5,5,4,1,2,3,3,4,5,5,4,5,7,8,7,
         - - - - - - - - - - - - - - - - - - - - - - - - - - -
A047679: 1,2,1,3,3,2,1,4,5,5,4,3,3,2,1,5,7,8,7,7,8,7,5,4,5,5,4,

(skip the first two terms from the start of A007305 as given in OEIS)

The fractions represented as prime factors, and mapped to polynomials:

1/1 -> (no factors) -> 0 polynomial.
1/2 -> 2^-1 -> -1 (constant polynomial).
2/1 -> 2^1  -> +1 (constant polynomial).
1/3 -> 3^-1 -> -x
2/3 -> 2^1 * 3^-1 -> -x + 1
3/2 -> 2^-1 * 3^1 -> x - 1
3/1 -> 3^1 -> x
1/4 -> 2^-2 -> -2 (constant polynomial)
2/5 -> 2^1 * 5^-1 -> -1(x^2) + 1
3/5 -> 3^1 * 5^-1 -> -1(x^2) + x
3/4 -> 3^1 * 2^-2 -> x - 2
4/3 -> 3^-1 * 2^2 -> -x + 2
5/3 -> 3^-1 * 5^1 -> x^2 - x
5/2 -> 2^-1 * 5^1 -> x^2 - 1 (generates A084099)
4/1 -> 2^2 -> 2

e.g. 5/18 = 5^1 * 3^-2 * 2^-1 would correspond to x^2 - 2x - 1
and 7/30 = 7^1 * 5^-1 * 3^-1 * 2^-1 would correspond to x^3 - x^2 - x - 1.
and 11/25 = p(5)^1 * p(3)^-2 would correspond to x^4 - 2x^2
(where p(n) is the nth prime).

Of course, one could/should skip the constant polynomials, in which
case one has to skip the powers of 2 (including negative exponents 1/2, 
1/4, 1/8, ...)
in the Stern-Brocot tree. These are in well-known places.

Then create a table, where each row n is the sequence generated by a said
polynomial. (By convention a zero-sequence if a constant polynomial?)

Thus all such sequences can be collected to one table, and thus linked
to each other via that "pivot".
(Well, we _knew_ that they were enumerable... There are other mappings 
as well, but maybe more cumbersome.)


Terveisin,

Antti



>
> -----Original Message-----
> From: Ed Pegg Jr <edp at wolfram.com>
>
> 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, 
> 96337632, 512384048, 
> 2725180256, 14494220608, 77089370720, 410009702400, 2180689172736, 
> 11598274968256, 
> 61686912523392, 328089753612800 
>  
> Table[RootSum[4+12*#1-6*#1^2-8*#1^3+#1^4+#1^5&, #^n&],{n,1,20}] 
>  
> {-1, 17, -7, 89, -51, 521, -379, 3201, -2851, 20137, -21275, 128369, 
> -156339, 825401, 
> -1131147, 5340993, -8075171, 34736777, -57020027, 226895889} 
>  
> Table[RootSum[4+4*#1-6*#1^2-#1^3+#1^4&, #^n&],{n,1,20}] 
>  
> {1,13,7,65,51,361,379,2081,2707,12233,18635,72881,124931,438745,822267,26
> 63489, 
> 5342387,16279273,34390571,100042705} 
>  
> Are there any polynomials that give some OEIS Sequences? As an aside, 
> those last two polynomials 
> are factors of the characteristic polynomial of the Balaban 11-cage. I 
> wondered if the two polynomials 
> were related in some strange way... and I then noticed they shared {1, 
> 7, 51, 379} in their sequences. 
>  
> Ed Pegg Jr. 
>
>
> ___________________________________________________
> Try the New Netscape Mail Today!
> Virtually Spam-Free | More Storage | Import Your Contact List
> http://mail.netscape.com
>
>