Continued fraction convergents to sqrt(3)

[creigh at o2online.de]

Hello,

(a(n)) and (b(n)), below, are two sequences I hope to be able to submit at 
some time over the weekend:
 
Conjecture:

Define a(n) =  A001835(n/2+1)_0, n even
                 =  A003699((n+3)/2)_1, n odd

 b(n) =   A054485(n/2)_0, n even
        =  -A052530((n+1)/2)_0, n odd

Then | a(n) + b(n) | = 2*A002531(n+1)

A002531, Numerators of continued fraction convergents to sqrt(3).
A052530, A (nice) simple regular expression 
A054485, A second order recursive sequence
A003699, Number of Hamilton cycles in C_4 X P_n
A001835  (nice)
 
Examples:
2*A002531(1) = 2 =  1 + 1 = | a(0) + b(0) | = A001835(1) + A054485(0)     
// n = 0
2*A002531(2) = 4 =  | 6 - 2 | = | a(1) + b(1) | =  A003699(2) - A052530
(1)  // n = 1
2*A002531(3) = 10 =  3 + 7 =   | a(2) + b(2) |    // n = 2
2*A002531(4) = 14 =  |22 - 8| =   | a(2) + b(2) |   // n = 3
2*A002531(5) = 38 =  11 + 27  =   | a(2) + b(2) |   // n = 4

(It looks as if the absolute value signs will eventually get dropped.)

Concering FAMP's results leading to the above formula:
The generating floretion (see "results" below) is surprisingly simple (compare 
this, for example, with the floretion given at the comment to http://www.
research.att.com/projects/OEIS?Anum=A001541
from yesterday). Also notice that "jes", "les", and "tes", below, do not 
lead to any new formula in this case (and that additional sequences appear 
which will also probably be of interest).  

p.s. One of my favorite sequences found so far is the sequence
4achuseq[J]: -4, -3, -5, -5, -6, -6, -6, -5, -3, 1, 8, 20, 40, 73, 127, 
215
which I found "buried" inside a batch of FAMP sequences (the sequence
Fibonacci numbers - 3 is in this same batch) 
 
Sincerely,
Creighton 

Results for the floretion:  

 *********************************************** 
  + 0.5'i + 0.5'k - 1i' + 0.5j' + 0.5k' 
 *********************************************** 

 
 ****************************  Static Symmetries  
 
 (squaring symmetries:) 
  
 vesseq: 1, -2, -5, 7, 19, -26, -71, 97, 265, -362, -989, 1351, 3691, -5042, 
-13775, 18817, 51409,   
 tesseq: 0, -2, 0, 7, 0, -26, 0, 97, 0, -362, 0, 1351, 0, -5042, 0, 18817, 
0, -70226, 0, 262087, 0,  
 lesseq: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 0 
 jesseq: 1, 0, -5, 0, 19, 0, -71, 0, 265, 0, -989, 0, 3691, 0, -13775, 
0, 51409, 0, -191861, 0, 716035,  
 
 identity: ves = jes + les + tes  
 
 (Emmy's Three, subgroup symmetries:) 
  
 2em[I]seq: -1, -4, 1, 14, -3, -52, 11, 194, -41, -724, 153, 2702, -571, -10084, 
2131, 37634 
 2em[I*]seq: 3, 0, -11, 0, 41, 0, -153, 0, 571, 0, -2131, 0, 7953, 0, -29681, 
0
 
 2em[J]seq: 1, -6, -3, 22, 11, -82, -41, 306, 153, -1142, -571, 4262, 2131, -15906, 
-7953, 59362 
// = (a(n))
 2em[J*]seq: 1, 2, -7, -8, 27, 30, -101, -112, 377, 418, -1407, -1560, 
5251, 5822, -19597, -21728
// = (b(n))

 em[K]seq: 1, -2, -4, 7, 15, -26, -56, 97, 209, -362, -780, 1351, 2911, -5042, 
-10864, 18817 
 em[K*]seq: 0, 0, -1, 0, 4, 0, -15, 0, 56, 0, -209, 0, 780, 0, -2911, 0
 
 2famseq: 0, -5, 0, 18, 0, -67, 0, 250, 0, -933, 0, 3482, 0, -12995, 0, 
48498
 2fam*seq: 2, 1, -10, -4, 38, 15, -142, -56, 530, 209, -1978, -780, 7382, 
2911, -27550, -10864 

 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