# Reply from On-Line Encyclopedia of Integer Sequences (fwd)

Richard Guy rkg at cpsc.ucalgary.ca
Fri Oct 26 17:23:49 CEST 2001

What I meant was the values of  n  for which length of
period of continued fraction for  sqrt(3n^2) was  2.

From the version below we see that this sequence is

1 2 3 4 7 11 14 15 26 28 41 52 56 79 98 ...

Is it connected with the convergents to the continued
fraction for  sqrt(3) ? :

0  1  1  2  5  7  19  26  71  97
-  -  -  -  -  -  --  --  --  --  ...
1  0  1  1  3  4  11  15  41  56

R.

On Fri, 26 Oct 2001, Meeussen Wouter (bkarnd) wrote:

> In[1]:=Table[Length[Last[ContinuedFraction[Sqrt[3] n]]],{n,128}]
> Out[1]=
> {2, 2, 2, 2, 4, 8, 2, 4, 10, 4, 2, 8, 6, 2, 2, 8, 6, 20, 4, 8, 14, 8, 8, 12,
> 16, 2, 30, 2, 16, 4, 18, 16, 10, 20, 16, 20, 18, 8, 6, 12, 2, 12, 8, 8, 6,
> 20, 20, 20, 30, 36, 6, 2, 8, 68, 14, 2, 16, 32, 22, 4, 38, 18, 40, 36, 6,
> 28,
> 10, 20, 40, 8, 4, 40, 18, 22, 16, 12, 28, 2, 46, 20, 98, 8, 46, 8, 6, 16,
> 16,
> 24, 50, 16, 16, 28, 50, 48, 12, 48, 2, 30, 50, 28, 16, 12, 46, 8, 12, 16,
> 50,
> 52, 66, 36, 14, 4, 62, 28, 32, 28, 16, 64, 36, 8, 74, 30, 10, 14, 84, 36,
> 62,
> 72}
>
> In[2]:=Table[Length[Last[ContinuedFraction[Sqrt[3^(2n+1)] ]]],{n,10}]
> Out[2]=
> {2, 10, 30, 98, 270, 818, 2382, 7282, 21818, 65650}
>
> I don't grok RKG's "-- it's the ranks of terms in the
> last seq that I sent which have value 2" in his last mail
>
>
> Wouter.
>
>
>
>
>
> -----Original Message-----
> From: Richard Guy [mailto:rkg at cpsc.ucalgary.ca]
> Sent: donderdag 25 oktober 2001 18:16
> To: ogerard at ext.jussieu.fr; seqfan at ext.jussieu.fr; Neil J. A. Sloane
> Subject: Reply from On-Line Encyclopedia of Integer Sequences (fwd)
>
>
> Here's a raft of new sequences, if anyone thinks
> they are worthwhile and has the patience to
> compute them. The one below is the period lengths
> of the continued fraction for sqrt(3n^2) for
> n = 1, 2, 3, ...
>
> Here are a few more terms:
>
> 2 2 2 2 4 8 2 4 10 4 2 8 6 2 2 8 6 20 4 8 14 8 8 12 16 2 28
>
> Similar sequences may or may not be of interest
> with 3 replaced by other squarefree numbers.
>
> Also some subsequences may be quite striking, e.g.
> for sqrt(3^(2k+1)) :  2 2 10 28 ... much bigger   R.
>
>
>
> ---------- Forwarded message ----------
> Date: Thu, 25 Oct 2001 12:42:24 -0400 (EDT)
> To: rkg at cpsc.ucalgary.ca
> Subject: Reply from On-Line Encyclopedia of Integer Sequences
>
> Matches (up to a limit of 50) found for  2 2 2 2 4 8 2 4 10 4 2 8 6 2 2 8 6
> 20 4 8  :
>
>
> Even though there are a large number of sequences in the table, at least
> one of yours is not there! Please send it to me using
> the submission form on the sequence web page
> http://www.research.att.com/~njas/sequences/Submit.html
> and I will (probably) add it!  Include a brief description. Thanks!
>
>
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