[seqfan] Re: In the vein of 103314
David Wilson
davidwwilson at comcast.net
Wed Dec 20 20:43:14 CET 2017
Pretty sure this is wrong.
There is 1 sum of 1 1st root of 1, namely {1}.
There are 3 sums of 2 2nd roots of 1, namely {-2, 0, 2} = {-1 + -1, -1 + 1, 1 + 1}
So your sequence should definitely start (1, 3, ...) and if I counted right (1, 3, 10, 25, ...)
The set of sums of n nth-roots includes 0 together with nonzero sums that exhibit n-fold symmetry about the origin of the complex plane.
We should therefore have
a(n) == 1 (mod n)
which your sequence below does not obey.
> -----Original Message-----
> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Wouter
> Meeussen
> Sent: Wednesday, December 20, 2017 12:58 PM
> To: Sequence Fanatics Discussion list
> Subject: [seqfan] Re: In the vein of 103314
>
> how many different absolute values can we get from the set of n-th roots of
> 1?
> I count (for n=1 .. 12)
>
> Table[Tally[Sort[Flatten[Table[dist/@ListNecklaces[n, Join[1 + 0*Range[i],
> 0*Range[n - i]],Dihedral], {i, 1,Floor[ n/2]}]]]]//Length,{n,12}]
>
> {0, 1, 1, 3, 3, 4, 8, 10, 17, 18, 62, 24}
>
> Wouter
>
>
> -----Original Message-----
> From: Max Alekseyev
> Sent: Wednesday, December 20, 2017 4:42 PM
> To: Sequence Fanatics Discussion list
> Subject: [seqfan] Re: In the vein of 103314
>
> They all must be distinct as otherwise the n-th primitive root would be a zero
> of a polynomial of degree n-1.
> Hence, there are binomial(k+n-1,n-1) distinct values of k-term sums of nth
> roots of 1, for any k>=1.
>
> Regards,
> Max
>
> On Tue, Dec 19, 2017 at 4:24 PM, David Wilson <davidwwilson at comcast.net>
> wrote:
>
> > How many distinct values are taken on by a sum of n nth roots of 1?
> >
> >
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