Note on Fib(n), L(n)

[creigh at o2online.de]

I find similarities among the following three formula worth noting: 
(for an explanation of notation, see below or http://www.crowdog.de -> 
Floretions;  a Java "FAMP" program given can also be downloaded from
there )

A057681(n)  = ves( (E'x)^n )/2 
E' = ( - i' - i' + 'ii' + 'jj' + 'kk' + 'jk' + kj' + 1 )/4
x = 'i + 'i + 'ji' + 'ki' + 1 (signs are preserved) 

A000032(n) = L(n) = ves( (Ex)^n ) (disregarding signs) 
E = ( i' + i' + 'ii' + 'jj' + 'kk' + 'jk' + kj' + 1 )/4
x = 'i + 'i + 'ji' + 'ki' + 1 

A000045(n) = Fib(n) = ves( (E''x)^n ) 
E'' = ( - i' - i' - 'ii' + 'jj' + 'kk' + 'jk' + kj' - 1 )/4
x = 'i + 'i + 'ji' + 'ki' + 1 

****************
The following is becoming part of a "standard procedure": 
once a formula for any sequence has been found (since forall z 
in "Floretion Algebra": jes(z) + les(z) + tes(z) = ves(z) by 
definition), "jes", "les", and "tes" can be used to separate (or "disect") 
it into the termwise sum of three sequences. This often produces 
interesting results (which have only begun to be documented)
Examples:
(1)  A001906(n+1) = (A005248(n+1) - A011783(n))/2   
(2)  A081010 a(n) = 2 + .5[A001906(n+1)]^2 + .5[A011783(n)]^2 

To arrive at the above, notice that ves( (Ex)^n ) = A001906(n+1) with
E = .25( - 'i - i' + 'ii' + 'jj' + 'kk' + 'jk' + 'kj' + 1 ) 
x = 'i + 'k + j' + 'jj' + 1 
(this was not a total "shot in the dark"- it was known that "E", 
along with a few other elements, have special "symmetric" properties)
 
It was then verified that 
jes( (Ex)^n ) = - A011783(n)/2 
les( (Ex)^n ) = tes( (Ex)^n ) =  A005248(n+1)/4 
ves( (Ex)^n ) = A001906(n+1) 

The first formula now follows immediately using 
jes(z) + les(z) + tes(z) = ves(z) 

The second formula follows from the first using the relation
A005248(n) = sqrt((A005248(n)^2-4)/5) given at 
http://www.research.att.com/projects/OEIS?Anum=A001906

**************
Explanation of Formula: 

For elements of the factor space Q x Q / {(1,1), (-1,-1)}, 
Q = Quaternions I write 'i to mean [ (i, 1) ] = [ (-i, -1) ], 
i' to mean [ (1, i) ] = [ (-1, -i) ], 'ij' to mean [ (i, j) ], etc. 
Let F be the algebra over the reals generated from this group. 
A typical element z of F can be written by 16 basis vectors:
x =    x(1) 'i + x(2) 'j + x(3) 'k + x(4) i' + x(5) j' + x(6) k' + x(7) 
'ii' 
      + x(8) 'jj' + x(9) 'kk'  + x(10)'ij' + x(11) 'ik' + x(12) 'ji' + 
x(13) 'jk'
      + x(14) 'ki' + x(15) 'kj' + x(16) 1

**********
Note: technically, one need only consider the ring F* generated 
by Q x Q / {(1,1), (-1,-1)}, however, we often need to consider elements
of the form x = tz, where z in F* and  t = 1/n  for some n in naturals; 
on occasion I've also needed to set n = sqrt(2). Regardless, one could
always deal exclusively with the ring and divide off these numbers 
afterwards, ex.:
A057681(n)  = ves( (E'x)^n )/2 
E' = ( - i' - i' + 'ii' + 'jj' + 'kk' + 'jk' + kj' + 1 )/4
x = 'i + 'i + 'ji' + 'ki' + 1 
becomes
A057681(n)  = ves( (E'''x)^n )/(2*4^n) 
E''' = ( - i' - i' + 'ii' + 'jj' + 'kk' + 'jk' + kj' + 1 )
x = 'i + 'i + 'ji' + 'ki' + 1 
***********

Define the (linear) functions: 
tes: F  ->  reals, 
ves: F ->  reals, 
jes: F  ->  reals,
les: F  ->  reals 

tes( (x_1)'i + (x_2)'j + ... + (x_16)1 ) =  x_16 
ves( (x_1)'i + (x_2)'j + ... + (x_16)1 ) = x_1 + x_2 + ... + x_16 
jes( (x_1)'i + (x_2)'j +  ... + (x_16)1 ) = x_1 + x_2 + x_3 + x_4 + x_5 
+ x_6 

les = ves - tes - jes

Sincerly,
Creighton Dement