# [seqfan] Re: In the vein of 103314

Wouter Meeussen wouter.meeussen at telenet.be
Wed Dec 20 18:58:24 CET 2017

```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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>

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```