Comments on: the iguanodon dissected http://www.neverendingbooks.org/index.php/the-inguanodon-dissected.html lieven le bruyn's blog Mon, 08 Mar 2010 18:45:28 +0100 http://wordpress.org/?v=2.9.1 hourly 1 By: neverendingbooks » Blog Archive » Farey symbols of sporadic groups http://www.neverendingbooks.org/index.php/the-inguanodon-dissected.html/comment-page-1#comment-5120 neverendingbooks » Blog Archive » Farey symbols of sporadic groups Thu, 20 Mar 2008 19:30:47 +0000 http://www.neverendingbooks.org/?p=42#comment-5120 <p>[...] the inguanodon post Ive added yet another construction of the Mathieu groups and starting from (half of) the Farey [...]</p> [...] the inguanodon post Ive added yet another construction of the Mathieu groups and starting from (half of) the Farey [...]

]]> By: more iguanodons via kfarey.sage at neverendingbooks http://www.neverendingbooks.org/index.php/the-inguanodon-dissected.html/comment-page-1#comment-3456 more iguanodons via kfarey.sage at neverendingbooks Tue, 11 Dec 2007 11:25:40 +0000 http://www.neverendingbooks.org/?p=42#comment-3456 <p>[...] what it is worth, Ive computed some more terms in the iguanodon series. Here they [...]</p> [...] what it is worth, Ive computed some more terms in the iguanodon series. Here they [...]

]]> By: Jonathan Vos Post http://www.neverendingbooks.org/index.php/the-inguanodon-dissected.html/comment-page-1#comment-3405 Jonathan Vos Post Wed, 28 Nov 2007 16:54:53 +0000 http://www.neverendingbooks.org/?p=42#comment-3405 <p>A133404 Table of sum of numerator and denominator of Farey sequences, read by rows.</p> <p>http://www.research.att.com/~njas/sequences/A133404</p> <p>1, 2, 1, 3, 2, 1, 4, 3, 5, 2, 1, 5, 4, 3, 5, 7, 1, 6, 5, 4, 7, 3, 8, 5, 7, 9, 2, 1, 7, 6, 5, 4, 7, 3, 8, 5, 7, 9, 11, 2, 1, 8, 7, 6, 5, 9, 4, 7, 10, 3, 11, 8, 5, 12, 7, 9, 11, 13, 2, 1, 9, 8, 7, 6, 5, 9, 4, 11, 7, 10, 3, 11, 8, 13, 5, 12, 7, 9, 11, 13, 15, 2 </p> <p>OFFSET<br /> 1,2</p> <p>COMMENT<br /> Start with the Farey sequence F(n) of order n which is the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size. Each row begins with the sum 1 from {0/1}. Each row ends with the sum 2 from {1/1}. The number of elements of the n-th row is A005728(n).</p> <p>FORMULA<br /> A007305/A007306 maps to A007305 A007306 as shown in examples.</p> <p>EXAMPLE </p> <p>F(1) = (0/1, 1/1) -> (0 1=1, 1 1=2).</p> <p>F(2) = (0/1, 1/2, 1/1) -> (0 1=1, 1 2=3, 1 1=2).</p> <p>F(3) = (0/1, 1/3, 1/2, 2/3, 1/1) -> (0 1=1, 1 3=4, 1 2=3, 2 3=5, 1 1=2).</p> <p>F(4) = (0/1, 1/4, 1/3, 1/2, 2/3, 3/4, 1/1) -> (0 1=1, 1 4=5, 1 3=4, 1 2=3, 2 3=5, 3 4=7, 1 1=2).</p> <p>The 5th row is formed from the 5th row of the table of Farey fractions:</p> <p>F(5) = (0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1) whose sum of numerators and denominators is (1, 6, 5, 4, 7, 3, 8, 5, 7, 9, 2).</p> <p>F(6) = (0/1, 1/6, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 5/6, 1/1} whose sums are (1, 7, 6, 5, 4, 7, 3, 8, 5, 7, 9, 11, 2).</p> <p>F(7) = (0/1, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 2/5, 3/7, 1/2, 4/7, 3/5, 2/3, 5/7, 3/4, 4/5, 5/6, 6/7, 1/1) whose sums are (1, 8, 7, 6, 5, 9, 4, 7, 10, 3, 11, 8, 5, 12, 7, 9, 11, 13, 2).</p> <p>F(8) = (0/1, 1/8, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 3/8, 2/5, 3/7, 1/2, 4/7, 3/5, 5/8, 2/3, 5/7, 3/4, 4/5, 5/6, 6/7, 7/8, 1/1) whose sums are (1, 9, 8, 7, 6, 5, 9, 4, 11, 7, 10, 3, 11, 8, 13, 5, 12, 7, 9, 11, 13, 15, 2).</p> <pre><code>CROSSREFS </code></pre> <p>Cf. A005728, A007305, A007306, A049448.</p> <p>KEYWORD<br /> easy,more,nonn,tabl,new</p> <p>AUTHOR </p> <p>Jonathan Vos Post (jvospost2(AT)yahoo.com), Nov 24 2007</p> A133404 Table of sum of numerator and denominator of Farey sequences, read by rows.

http://www.research.att.com/~njas/sequences/A133404

1, 2, 1, 3, 2, 1, 4, 3, 5, 2, 1, 5, 4, 3, 5, 7, 1, 6, 5, 4, 7, 3, 8, 5, 7, 9, 2, 1, 7, 6, 5, 4, 7, 3, 8, 5, 7, 9, 11, 2, 1, 8, 7, 6, 5, 9, 4, 7, 10, 3, 11, 8, 5, 12, 7, 9, 11, 13, 2, 1, 9, 8, 7, 6, 5, 9, 4, 11, 7, 10, 3, 11, 8, 13, 5, 12, 7, 9, 11, 13, 15, 2

OFFSET
1,2

COMMENT
Start with the Farey sequence F(n) of order n which is the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size. Each row begins with the sum 1 from {0/1}. Each row ends with the sum 2 from {1/1}. The number of elements of the n-th row is A005728(n).

FORMULA
A007305/A007306 maps to A007305 A007306 as shown in examples.

EXAMPLE

F(1) = (0/1, 1/1) -> (0 1=1, 1 1=2).

F(2) = (0/1, 1/2, 1/1) -> (0 1=1, 1 2=3, 1 1=2).

F(3) = (0/1, 1/3, 1/2, 2/3, 1/1) -> (0 1=1, 1 3=4, 1 2=3, 2 3=5, 1 1=2).

F(4) = (0/1, 1/4, 1/3, 1/2, 2/3, 3/4, 1/1) -> (0 1=1, 1 4=5, 1 3=4, 1 2=3, 2 3=5, 3 4=7, 1 1=2).

The 5th row is formed from the 5th row of the table of Farey fractions:

F(5) = (0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1) whose sum of numerators and denominators is (1, 6, 5, 4, 7, 3, 8, 5, 7, 9, 2).

F(6) = (0/1, 1/6, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 5/6, 1/1} whose sums are (1, 7, 6, 5, 4, 7, 3, 8, 5, 7, 9, 11, 2).

F(7) = (0/1, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 2/5, 3/7, 1/2, 4/7, 3/5, 2/3, 5/7, 3/4, 4/5, 5/6, 6/7, 1/1) whose sums are (1, 8, 7, 6, 5, 9, 4, 7, 10, 3, 11, 8, 5, 12, 7, 9, 11, 13, 2).

F(8) = (0/1, 1/8, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 3/8, 2/5, 3/7, 1/2, 4/7, 3/5, 5/8, 2/3, 5/7, 3/4, 4/5, 5/6, 6/7, 7/8, 1/1) whose sums are (1, 9, 8, 7, 6, 5, 9, 4, 11, 7, 10, 3, 11, 8, 13, 5, 12, 7, 9, 11, 13, 15, 2).

CROSSREFS

Cf. A005728, A007305, A007306, A049448.

KEYWORD
easy,more,nonn,tabl,new

AUTHOR

Jonathan Vos Post (jvospost2(AT)yahoo.com), Nov 24 2007

]]> By: lieven http://www.neverendingbooks.org/index.php/the-inguanodon-dissected.html/comment-page-1#comment-3361 lieven Fri, 16 Nov 2007 13:50:55 +0000 http://www.neverendingbooks.org/?p=42#comment-3361 <p>Apologies for using ambiguous terminology. I should have used 'Farey symbol' as in Kulkarni's paper. nothing to do with codes as such as far as i know.</p> Apologies for using ambiguous terminology. I should have used ‘Farey symbol’ as in Kulkarni’s paper. nothing to do with codes as such as far as i know.

]]> By: Doug http://www.neverendingbooks.org/index.php/the-inguanodon-dissected.html/comment-page-1#comment-3355 Doug Fri, 16 Nov 2007 02:50:18 +0000 http://www.neverendingbooks.org/?p=42#comment-3355 <p>Please delete comment of Nov 16th, 2007 at 3:41 am.</p> <p>If possible, consider replacing appropriate section of comment Nov 16th, 2007 at 3:35 am with:</p> <p>[A redo on:]</p> <p>Look at F(9) in this format: 0/1<br /> 1/9 2/9 #/# 4/9#/# 5/9 #/# 7/9 #/# 8/9 1/8 #/# 3/8 #/# 5/8 #/# 7/8 1/7 2/7 3/7 4/7 5/7 6/7<br /> 1/6 #/# #/# #/# 5/6<br /> 1/5 2/5 3/5 4/5<br /> 1/4 #/# 3/4<br /> 1/3 2/3<br /> 1/2<br /> 1/1</p> Please delete comment of Nov 16th, 2007 at 3:41 am.

If possible, consider replacing appropriate section of comment Nov 16th, 2007 at 3:35 am with:

[A redo on:]

Look at F(9) in this format: 0/1
1/9 2/9 #/# 4/9#/# 5/9 #/# 7/9 #/# 8/9 1/8 #/# 3/8 #/# 5/8 #/# 7/8 1/7 2/7 3/7 4/7 5/7 6/7
1/6 #/# #/# #/# 5/6
1/5 2/5 3/5 4/5
1/4 #/# 3/4
1/3 2/3
1/2
1/1

]]> By: Doug http://www.neverendingbooks.org/index.php/the-inguanodon-dissected.html/comment-page-1#comment-3354 Doug Fri, 16 Nov 2007 02:41:52 +0000 http://www.neverendingbooks.org/?p=42#comment-3354 <p>A redo on:</p> <p>Look at F(9) in this format: 0/1<br /> 1/9 2/9 ___ 4/9 5/9 ___ 7/9 ___ 8/9 1/8 ___ 3/8 ___ 5/8 ___ 7/8 1/7 2/7 3/7 4/7 5/7 6/7<br /> 1/6 ___ ___ ___ 5/6<br /> 1/5 2/5 3/5 4/5<br /> 1/4 ___ 3/4<br /> 1/3 2/3<br /> 1/2<br /> 1/1</p> A redo on:

Look at F(9) in this format: 0/1
1/9 2/9 ___ 4/9 5/9 ___ 7/9 ___ 8/9 1/8 ___ 3/8 ___ 5/8 ___ 7/8 1/7 2/7 3/7 4/7 5/7 6/7
1/6 ___ ___ ___ 5/6
1/5 2/5 3/5 4/5
1/4 ___ 3/4
1/3 2/3
1/2
1/1

]]> By: Doug http://www.neverendingbooks.org/index.php/the-inguanodon-dissected.html/comment-page-1#comment-3353 Doug Fri, 16 Nov 2007 02:35:15 +0000 http://www.neverendingbooks.org/?p=42#comment-3353 <p>Hi Lieven, </p> <p>Look at F(9) in this format: 0/1<br /> 1/9 2/9 4/9 5/9 7/9 8/9 1/8 3/8 5/8 7/8 1/7 2/7 3/7 4/7 5/7 6/7<br /> 1/6 5/6<br /> 1/5 2/5 3/5 4/5<br /> 1/4 3/4<br /> 1/3 2/3<br /> 1/2<br /> 1/1 </p> <p>Note the number of elements in: F(1) ~ 2 F(2) ~ 3 F(3) ~ 5 F(4) ~ 7 F(5) ~ 11 F(6) ~ 13 F(7) ~ 19 F(8) ~ 23 F(9) ~ 29 are Supersingular Primes, missing 17 [and of course 31, 41, 47, 59, 71] Coincidence?</p> <p>Is there an F(0)={0} [seems most likely] or the null set ={}?</p> <p>Thank you for modifying your blog into this more readable format [for me] that also allows me to view the entire comments of others.</p> Hi Lieven,

Look at F(9) in this format: 0/1
1/9 2/9 4/9 5/9 7/9 8/9 1/8 3/8 5/8 7/8 1/7 2/7 3/7 4/7 5/7 6/7
1/6 5/6
1/5 2/5 3/5 4/5
1/4 3/4
1/3 2/3
1/2
1/1

Note the number of elements in: F(1) ~ 2 F(2) ~ 3 F(3) ~ 5 F(4) ~ 7 F(5) ~ 11 F(6) ~ 13 F(7) ~ 19 F(8) ~ 23 F(9) ~ 29 are Supersingular Primes, missing 17 [and of course 31, 41, 47, 59, 71] Coincidence?

Is there an F(0)={0} [seems most likely] or the null set ={}?

Thank you for modifying your blog into this more readable format [for me] that also allows me to view the entire comments of others.

]]> By: Kea http://www.neverendingbooks.org/index.php/the-inguanodon-dissected.html/comment-page-1#comment-3352 Kea Fri, 16 Nov 2007 02:10:13 +0000 http://www.neverendingbooks.org/?p=42#comment-3352 <p>I googled 'Farey code' and got very few hits. One hit was a recent paper by Carpi and de Luca on languages (Europ. J. Comb. 28 (2007) 371-402), which looks interesting. Did these people define a Farey code? I was also wondering if you had further references like this one that might help me connect a few dots.</p> I googled ‘Farey code’ and got very few hits. One hit was a recent paper by Carpi and de Luca on languages (Europ. J. Comb. 28 (2007) 371-402), which looks interesting. Did these people define a Farey code? I was also wondering if you had further references like this one that might help me connect a few dots.

]]> By: Doug http://www.neverendingbooks.org/index.php/the-inguanodon-dissected.html/comment-page-1#comment-3306 Doug Sat, 10 Nov 2007 02:26:14 +0000 http://www.neverendingbooks.org/?p=42#comment-3306 <p>Look at F(9) in this format: 0/1<br /> 1/9 2/9 4/9 5/9 7/9 8/9 1/8 3/8 5/8 7/8 1/7 2/7 3/7 4/7 5/7 6/7<br /> 1/6 5/6<br /> 1/5 2/5 3/5 4/5<br /> 1/4 3/4<br /> 1/3 2/3<br /> 1/2<br /> 1/1 </p> <p>Note the number of elements in: F(1) ~ 2 F(2) ~ 3 F(3) ~ 5 F(4) ~ 7 F(5) ~ 11 F(6) ~ 13 F(7) ~ 19 F(8) ~ 23 F(9) ~ 29 are Supersingular Primes, missing 17 [and of course 31, 41, 47, 59, 71] Coincidence?</p> <p>Is there an F(0)={0} or the null set?</p> Look at F(9) in this format: 0/1
1/9 2/9 4/9 5/9 7/9 8/9 1/8 3/8 5/8 7/8 1/7 2/7 3/7 4/7 5/7 6/7
1/6 5/6
1/5 2/5 3/5 4/5
1/4 3/4
1/3 2/3
1/2
1/1

Note the number of elements in: F(1) ~ 2 F(2) ~ 3 F(3) ~ 5 F(4) ~ 7 F(5) ~ 11 F(6) ~ 13 F(7) ~ 19 F(8) ~ 23 F(9) ~ 29 are Supersingular Primes, missing 17 [and of course 31, 41, 47, 59, 71] Coincidence?

Is there an F(0)={0} or the null set?

]]>