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

David W. Wilson wilson at aprisma.com
Fri Oct 26 15:10:00 CEST 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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%I A000000
%S A000000
1,2,2,4,1,2,4,4,10,2,10,2,5,4,8,12,2,6,14,8,8,6,16,4,13,10,26,8,1,8,24,
%T A000000
24,6,2,2,6,15,18,16,20,4,8,30,6,10,16,28,8,36,22,2,20,19,30,10,12,14,2,
%U A000000 14,8,25,20,4,48,13,6,54,2,32,2,52,16,30,26,40,10,6,16,20,36,78,4,58,8
%N A000000 Period of continued fraction of sqrt(2)*n
%K A000000 nonn
%O A000000 1,2
%A A000000 rkg

%I A000001
%S A000001
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,
%T A000001
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,
%U A000001
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
%N A000001 Period of continued fraction of sqrt(3)*n
%K A000001 nonn
%O A000001 1,1
%A A000001 rkg

%I A000002
%S A000002
1,2,6,2,5,4,10,4,2,14,12,4,5,10,28,8,1,2,4,14,6,8,6,4,31,14,10,12,12,20,
%T A000002
8,20,20,2,52,2,19,4,28,24,18,8,50,12,28,6,10,4,70,62,8,18,7,10,6,8,8,12,
%U A000002
72,20,3,12,8,36,41,28,86,2,6,44,84,2,43,42,120,4,52,36,28,44,38,18,34,8
%N A000002 Period of continued fraction of sqrt(5)*n
%K A000002 nonn
%O A000002 1,2
%A A000002 rkg

%I A000003
%S A000003
2,2,6,4,2,6,6,8,2,2,4,12,16,6,12,12,22,2,18,2,24,4,20,20,18,12,10,8,22,
%T A000003
8,26,32,4,14,8,8,40,10,40,4,34,16,38,8,8,16,40,44,2,14,10,24,50,10,12,4,
%U A000003
18,22,22,8,56,26,20,60,32,4,58,24,60,4,68,20,34,40,58,24,28,44,70,8,50
%N A000003 Period of continued fraction of sqrt(6)*n
%K A000003 nonn
%O A000003 1,1
%A A000003 rkg

%I A000004
%S A000004
4,4,2,6,6,4,16,2,6,16,10,4,14,32,6,2,4,16,14,12,24,10,14,2,26,24,22,36,
%T A000004
38,12,24,8,14,8,52,8,34,32,14,16,52,32,46,14,2,14,52,2,120,64,4,32,50,
%U A000004 44,12,32,18,34,58,12,70,52,48,12,50,18,66,8,14,96,18,8,64,42,30,36,96
%N A000004 Period of continued fraction of sqrt(7)*n
%K A000004 nonn
%O A000004 1,1
%A A000004 rkg

%I A000005
%S A000005
1,2,2,8,7,2,10,12,8,18,18,4,4,10,14,24,11,8,2,28,12,18,38,8,39,8,32,12,
%T A000005
23,14,44,48,18,26,60,12,1,2,4,60,30,12,60,18,38,34,74,20,86,74,30,20,36,
%U A000005
28,96,8,2,38,30,28,45,44,28,100,54,24,36,44,34,60,104,28,47,2,74,2,36,8
%N A000005 Period of continued fraction of sqrt(10)*n
%K A000005 nonn
%O A000005 1,2
%A A000005 rkg

%I A000006
%S A000006
2,8,2,4,6,4,8,8,8,2,26,4,20,16,4,16,26,12,2,2,8,52,26,8,26,40,28,12,28,
%T A000006
2,38,40,38,40,12,12,46,8,20,8,46,12,50,64,8,30,58,12,52,18,22,40,62,40,
%U A000006 60,32,2,68,26,2,72,42,4,80,40,64,46,48,34,8,22,20,90,34,22,8,90,32,96
%N A000006 Period of continued fraction of sqrt(11)*n
%K A000006 nonn
%O A000006 1,1
%A A000006 rkg

%I A000007
%S A000007
5,6,6,6,1,8,22,12,4,2,10,4,35,20,2,24,22,4,58,2,24,10,60,8,11,74,12,24,
%T A000007
19,2,98,48,6,26,24,4,43,58,86,8,17,16,32,12,2,64,42,16,174,26,22,90,68,
%U A000007 12,12,20,62,30,10,2,20,98,24,92,47,8,206,26,56,28,70,8,93,98,26,58,20
%N A000007 Period of continued fraction of sqrt(13)*n
%K A000007 nonn
%O A000007 1,1
%A A000007 rkg

%I A000008
%S A000008
4,2,4,2,6,4,16,4,14,4,10,4,18,28,2,8,28,14,22,4,40,10,10,4,26,18,50,24,
%T A000008
6,2,4,20,32,28,40,16,50,30,14,4,20,40,54,18,10,14,52,8,136,22,48,20,70,
%U A000008
54,32,44,30,2,78,2,22,4,128,48,14,28,82,28,14,36,22,12,112,58,26,34,92
%N A000008 Period of continued fraction of sqrt(14)*n
%K A000008 nonn
%O A000008 1,1
%A A000008 rkg

%I A000009
%S A000009
2,4,8,2,10,8,2,2,4,20,10,8,12,8,26,4,14,8,22,16,2,12,18,8,42,24,12,8,26,
%T A000009
48,2,8,26,10,26,8,28,14,32,16,40,8,8,16,24,18,30,12,34,88,46,32,16,20,2,
%U A000009 8,50,52,46,48,6,2,2,20,64,52,56,10,14,40,4,8,78,64,130,18,22,64,26,28
%N A000009 Period of continued fraction of sqrt(15)*n
%K A000009 nonn
%O A000009 1,1
%A A000009 rkg

%I A000010
%S A000010
1,2,8,2,7,8,14,2,18,6,6,4,5,20,20,4,25,22,32,6,16,6,42,4,25,6,66,12,29,
%T A000010
20,58,8,2,58,40,16,39,36,16,6,27,12,68,4,24,42,80,4,94,50,112,6,48,58,
%U A000010
20,12,60,50,104,16,61,66,32,20,1,2,4,82,42,40,130,16,55,70,108,32,8,20
%N A000010 Period of continued fraction of sqrt(17)*n
%K A000010 nonn
%O A000010 1,2

%I A000011
%S A000011
6,12,2,12,10,8,22,20,10,10,12,8,4,12,12,40,6,28,90,10,24,28,22,24,46,4,
%T A000011
46,12,72,8,8,72,14,6,20,24,92,172,2,12,94,20,50,28,24,18,130,32,134,38,
%U A000011
6,8,140,84,32,16,102,152,130,8,134,24,60,160,4,24,146,6,18,12,172,52,28
%N A000011 Period of continued fraction of sqrt(19)*n
%K A000011 nonn
%O A000011 1,1
%A A000011 rkg

%I A000012
%S A000012
6,2,4,2,6,2,26,8,8,8,6,2,26,30,8,12,34,8,10,8,30,6,10,4,42,22,36,22,12,
%T A000012
4,26,28,4,26,60,8,8,10,38,4,78,26,22,8,20,14,80,8,202,38,34,26,32,32,2,
%U A000012
60,14,16,116,4,98,30,86,64,60,4,42,32,10,48,42,12,158,4,34,18,52,26,46
%N A000012 Period of continued fraction of sqrt(21)*n
%K A000012 nonn
%O A000012 1,1
%A A000012 rkg

%I A000013
%S A000013
6,6,2,8,16,2,4,20,10,16,54,8,34,4,14,40,46,10,16,32,2,62,50,20,68,30,46,
%T A000013
4,66,14,74,84,66,38,8,24,96,28,28,56,98,2,62,108,10,54,26,36,32,72,50,
%U A000013
26,128,38,158,12,28,70,138,28,146,82,6,156,32,58,150,100,66,8,54,52,198
%N A000013 Period of continued fraction of sqrt(22)*n
%K A000013 nonn
%O A000013 1,1
%A A000013 rkg

%I A000014
%S A000014
4,8,8,4,2,6,8,6,14,4,18,6,18,16,4,6,28,14,34,4,16,32,72,2,14,18,58,20,
%T A000014
42,4,22,8,40,56,8,18,54,64,14,4,18,20,66,20,16,144,6,2,4,28,60,22,74,54,
%U A000014
14,16,64,38,14,4,106,26,20,16,20,36,26,72,152,16,50,10,28,124,28,72,48
%N A000014 Period of continued fraction of sqrt(23)*n
%K A000014 nonn
%O A000014 1,1
%A A000014 rkg

```