# [seqfan] Re: In the vein of 103314 (castigating a defunct equus)

Wouter Meeussen wouter.meeussen at telenet.be
Thu Dec 21 17:24:25 CET 2017

```how many different absolute values do we get from the set of n-th roots of
1?

ahum, I made two errors :
1)  I should have counted from 0 and not from 1 in the iterator  {i,
1,Floor[ n/2]}
and
2) I should have counted from 2 and not from 1 in the iterator  {n, 1, 12}
(because 'ListNecklaces' burps on single-bead argument)

so I correct my previous muttering to n= (1) .. 16
(2), 2, 2, 3, 4, 4, 9, 10, 17, 18, 63, 24, 187, 96, 281, 241

example: for n=6 we have the following vectors and absolute values:

{0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 1}, {0, 1, 0, 1, 0, 1}  -> 0
{0, 0, 0, 1, 0, 1}, {0, 0, 1, 0, 1, 1}, {1, 0, 0, 0, 0, 0}  -> 1
{1, 1, 0, 0, 0, 0}   ->  sqrt(3)
{1, 1, 1, 0, 0, 0}   ->  2

remark that these binary vectors are not only bracelets, but also have a 0
<-> 1 symmetry : it's literaly a zero-sum game.

---------------
dist[li_]:=Round[10^23 Abs[Plus@@N[li Exp[2 Pi I (li
Range[Length[li]])/Length[li]],24]]];

Table[Tally[Sort[Flatten[Table[dist/@ListNecklaces[n, Join[1 + 0*Range[i],
0*Range[n - i]],Dihedral], {i, 0,Floor[ n/2]}]]]]//Length,{n,2,16}]
---------------

Wouter.

-----Original Message-----
From: Wouter Meeussen
Sent: Wednesday, December 20, 2017 6:58 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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> Seqfan Mailing list - http://list.seqfan.eu/
>

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```