An addition to last message

[creigh at o2online.de]

Using the method described in my last message, the following 
was found today:

A025170(n) = 
( A088137(n+1) )^2 + ( (A087455(n+1)/2 )^2 - ( A087455(n+2)/2 )^2

http://www.research.att.com/projects/OEIS?Anum=A025170
(Reciprocal Chebyshev polynomial of second kind evaluated at 3)
http://www.research.att.com/projects/OEIS?Anum=A088137
(Generalized Gaussian Fibonacci integers)
http://www.research.att.com/projects/OEIS?Anum=A087455
(Type 2 generalized Gaussian Fibonacci integers)

Ex. (n = 7):  559 = 13^2 + ( 43/2 )^2 - (17/2 )^2 = 
= 169 + 462,25 - 72,25  

See also "quaternions and perfect squares" at  
http://mathforum.org/discuss/sci.math/t/622432

The last message had slight errors in the first lines
(these would have made a big difference in the final result).
My apologies, it should have read:

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 
 
Sincerely,
Creighton