[SeqFan] Re: Suggestion for a sequence: weights on a circle, roots of unity and Chebyshev's polynomials.

[Eric W. Weisstein]

On Mon, 1 May 2006, Antti Karttunen wrote:

> I just found this:
>
> http://www.research.att.com/~njas/sequences/A113401
>
> A113401 Algebraic degree of Cos[2Pi/n] for constructible n-gons 
> (A003401).  1, 1, 1, 1, 2, 1, 2, 2, 2, 4, 4, 8, 4, 4, 4, 8, 8, 8, 8, 16, 
> 8, 16, 16, 16, 32, 16, 16, 16, 32, 32, 32, 32, 32, 32, 32, 64, 64, 128, 
> 64, 64, 64, 64, 64, 64, 64, 128, 128, 128, 128, 128, 128, 256, 128, 128, 
> 128, 256, 256, 256, 256, 512, 256, 256, 256, 256, 256, 256, 512
>
> OFFSET 1,5
> KEYWORD nonn,easy,nice
> AUTHOR E. W. Weisstein (eric(AT)weisstein.com), Oct 28, 2005
>
> I just wonder, isn't this possible at all to compute/estimate for 
> non-constructible polygons (heptagon, 9-gon, etc.)? (Some holes in my 
> higher algebra, I guess.) From Weisstein's web page: "The angles m*Pi/n 
> (with m,n integers) for which the trigonometric functions may be 
> expressed in terms of finite root extraction of real numbers are limited 
> to values of m which are precisely those which produce constructible 
> polygons." Still I wonder, how much we can say about the vertices of 
> heptagon (and their algebraic degree) just by using cos(2Pi/7) as x, and 
> then applying Chebyshev polynomials (or some other trick).

Isn't this just
http://www.research.att.com/~njas/sequences/A023022
?

See also http://mathworld.wolfram.com/TrigonometryAngles.html

Cheers,
-Eric