Sequences involving continued fraction convergents to sqrt(5)

Sun Oct 31 11:05:31 CET 2004

Liebe Folgfans, (why not once in German?)

Perhaps there is a person who would enjoy researching the following data- 
there are too many unlisted sequences and I do not have the time to submit 
these individually at the moment.

If you do the research (and are the first to submit) I consider it to be 
your sequence (though it would be nice if "FAMP" were given as the program 
used).  

Something interesting I notice is 
 vesseq(I*B)(n) = A001076(n+1) 
A001076 Denominators of continued fraction convergents to sqrt(5)

while  - vesseq(B*I)(n) = A097924(n) 
http://www.research.att.com/projects/OEIS?Anum=A097924

All other sequences (or at least the ones I can remember checking) are unlisted. 
By the way, tesseq(B*I) = tesseq(I*B) is not a special property of this 
case but is an identity which holds for any two floretions.   

Thanks!

Sincerely, 
Creighton 

Results for the floretion:  I*B
 **********************************************************
  + 1'k + 1k' - 1.5'ii' + 1'jj' - 1'kk' + 1'ij' + 1'ji' + 0.5'jk' + 0.5'kj' 
+ 0.5e 
 **********************************************************
I = .5 'i + .5 i' + .5 'jj' + .5 'kk' + 
B = -1 'i + 1 'j + 1 i' + 1 j' + 1 'kk' + 1 'ik' + 1 'jk' - 1 'ki' - 1 
'kj' + 

Invariant to swap operator: yes
I commutes with B: no

 ****************************  Static Symmetries ****** 

 (squaring symmetries:) 

 vesseq: 4, 17, 72, 305, 1292, 5473, 23184, 98209, 416020, 1762289 
 2tesseq: 1, 10, 37, 162, 681, 2890, 12237, 51842, 219601, 930250, 3940597 
 2lesseq: 3, 16, 63, 272, 1147, 4864, 20599, 87264, 369651, 1565872, 6633135 
 jesseq: 2, 4, 22, 88, 378, 1596, 6766, 28656, 121394, 514228, 2178310 

 identity: ves = jes + les + tes  

 (Emmy's Three, subgroup symmetries:) 

 em[I]seq: 0, 5, 16, 73, 304, 1293, 5472, 23185, 98208, 416021, 1762288 
 em[I*]seq: 4, 12, 56, 232, 988, 4180, 17712, 75024, 317812, 1346268, 5702888

 em[J]seq: 3, 13, 55, 233, 987, 4181, 17711, 75025, 317811, 1346269 
 em[J*]seq: 1, 4, 17, 72, 305, 1292, 5473, 23184, 98209, 416020

 em[K]seq: -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1 
 em[K*]seq: 5, 16, 73, 304, 1293, 5472, 23185, 98208, 416021, 1762288, 
7465177

 famseq: -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1
 fam*seq: 5, 16, 73, 304, 1293, 5472, 23185, 98208, 416021, 1762288, 7465177 

 identities: em[I] + em[J] + em[K] = 2*fam + ves  
 em[I] + em[I*] = em[J] + em[J*] = em[K] + em[K*] = fam + fam* = ves  

 (Chu's group, subgroup symmetries, incomplete:) 

 chuseq[J]: 6, 20, 90, 376, 1598, 6764, 28658, 121392, 514230, 2178308 
 (signed)chuseq[J]: 6, 20, 90, 376, 1598, 6764, 28658, 121392, 514230, 
2178308 
 2chutesseq[J]: 13, 50, 217, 914, 3877, 16418, 69553, 294626, 1248061, 5286866 
 2achuseq[J]: -5, -16, -73, -304, -1293, -5472, -23185, -98208, -416021, 
-1762288, -7465177 
 achu[J]+tesseq: -2, -3, -18, -71, -306, -1291, -5474, -23183, -98210, 
-416019 
 achu[K]+tesseq: 4, 17, 72, 305, 1292, 5473, 23184, 98209, 416020, 1762289 
 2achu[I]+tesseq: 3, 18, 71, 306, 1291, 5474, 23183, 98210, 416019, 1762290 

 identity: ves = chu[X] + achu[X] + tes

 ****************************  Dynamic Symmetries  ****

 2posseq: 13, 54, 229, 970, 4109, 17406, 73733, 312338, 1323085, 5604678 
 2negseq: -5, -20, -85, -360, -1525, -6460, -27365, -115920, -491045, -2080100, 
-8811445 

 identity: pos + neq = ves  

 Results for the floretion:  B*I
 ******************************************************** 
  - 1'j - 1j' - 1.5'ii' - 1'jj' + 1'kk' + 1'ik' - 0.5'jk' + 1'ki' - 0.5'kj' 
+ 0.5e 
 **********************************************************

 ****************************  Static Symmetries  

 (squaring symmetries:) 

 vesseq: -2, -7, -30, -127, -538, -2279, -9654, -40895, -173234, -733831 
 2tesseq: 1, 10, 37, 162, 681, 2890, 12237, 51842, 219601, 930250, 3940597 
 2lesseq: -1, -8, -29, -128, -537, -2280, -9653, -40896, -173233, -733832, 
-3108557 
 jesseq: -2, -8, -34, -144, -610, -2584, -10946, -46368, -196418, -832040, 
-3524578 

 identity: ves = jes + les + tes  

 (Emmy's Three, subgroup symmetries:) 

 em[I]seq: -2, -3, -18, -71, -306, -1291, -5474, -23183, -98210, -416019, -1762290 
 em[I*]seq: 0, -4, -12, -56, -232, -988, -4180, -17712, -75024, -317812, 
-1346268

 em[J]seq: -1, -3, -13, -55, -233, -987, -4181, -17711, -75025, -317811, -1346269 
 em[J*]seq: -1, -4, -17, -72, -305, -1292, -5473, -23184, -98209, -416020, 
-1762289

 em[K]seq: -1, 1, -1, 1, -1, 1, -1, 1, -1, 1 
 em[K*]seq: -1, -8, -29, -128, -537, -2280, -9653, -40896, -173233, -733832

 famseq: -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1
 fam*seq: -1, -8, -29, -128, -537, -2280, -9653, -40896, -173233, -733832, 
-3108557 

 identities: em[I] + em[J] + em[K] = 2*fam + ves  
 em[I] + em[I*] = em[J] + em[J*] = em[K] + em[K*] = fam + fam* = ves  

 (Chu's group, subgroup symmetries, incomplete:) 

 chuseq[J]: -2, -12, -46, -200, -842, -3572, -15126, -64080, -271442, -1149852, 
-4870846 
 (signed)chuseq[J]: -2, -12, -46, -200, -842, -3572, -15126, -64080, -271442, 
-1149852, -4870846 
 2chutesseq[J]: -3, -14, -55, -238, -1003, -4254, -18015, -76318, -323283, 
-1369454, -5801095 
 2achuseq[J]: -1, 0, -5, -16, -73, -304, -1293, -5472, -23185, -98208, 
-416021 
 achu[J]+tesseq: 0, 5, 16, 73, 304, 1293, 5472, 23185, 98208, 416021 
 achu[K]+tesseq: -2, -7, -30, -127, -538, -2279, -9654, -40895, -173234, 
-733831 
 2achu[I]+tesseq: -1, 2, 3, 18, 71, 306, 1291, 5474, 23183, 98210 

 identity: ves = chu[X] + achu[X] + tes