RootSum Sequences

[franktaw at netscape.net]

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


-----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. 


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