Cayley-Dickson construction

[Paul D. Hanna]

Joerg, and Seqfans, 
    A very interesting topic. 
> ... for hypercomplex numbers. 
> corresponding sequence(s) do not seem exist. 
> There is only id:A096809 which could use much clarification. 
 
Yes; I will try (when I catch my breath) to clarify this table. 
It was my attempt in my early teenage years at finding a generalization 
to Hamilton's quaternions, without knowledge of current theory. 
I submitted it 2 years ago as an example of binary fractal sequences. 
  
 
I have a question in regard to your CD table of signs:
> 
>              0 1 2 3 4 5 6 7  8 9 a b c d e f
> 
>        0:    + + + + + + + +  + + + + + + + +
>        1:    + - - + - + + -  - + + - + - - +
>        2:    + + - - - - + +  - - + + + + - -
>        3:    + - + - - + - +  - + - + + - + -
>        4:    + + + + - - - -  - - - - + + + +
>        5:    + - + - + - + -  - + - + - + - +
>        6:    + - - + + - - +  - + + - - + + -
>        7:    + + - - + + - -  - - + + - - + +
> 
>        8:    + + + + + + + +  - - - - - - - -
>        9:    + - + - + - - +  + - + - + - - +
>        a:    + - - + + + - -  + - - + + + - -
>        b:    + + - - + - + -  + + - - + - + -
>        c:    + - - - - + + +  + - - - - + + +
>        d:    + + - + - - - +  + + - + - - - +
>        e:    + + + - - + - -  + + + - - + - -
>        f:    + - + + - - + -  + - + + - - + -
> 
 
Would there be any advantage to submitting a table T 
of exponents of -1 such that  CD(n,k) = (-1)^T(n,k) 
like as the table below? 
 
0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0, 
0,1,1,0,1,0,0,1, 1,0,0,1,0,1,1,0, 
0,0,1,1,1,1,0,0, 1,1,0,0,0,0,1,1, 
0,1,0,1,1,0,1,0, 1,0,1,0,0,1,0,1, 
0,0,0,0,1,1,1,1, 1,1,1,1,0,0,0,0, 
0,1,0,1,0,1,0,1, 1,0,1,0,1,0,1,0, 
0,1,1,0,0,1,1,0, 1,0,0,1,1,0,0,1, 
0,0,1,1,0,0,1,1, 1,1,0,0,1,1,0,0, 
 
0,0,0,0,0,0,0,0, 1,1,1,1,1,1,1,1, 
0,1,0,1,0,1,1,0, 0,1,0,1,0,1,1,0, 
0,1,1,0,0,0,1,1, 0,1,1,0,0,0,1,1, 
0,0,1,1,0,1,0,1, 0,0,1,1,0,1,0,1, 
0,1,1,1,1,0,0,0, 0,1,1,1,1,0,0,0, 
0,0,1,0,1,1,1,0, 0,0,1,0,1,1,1,0, 
0,0,0,1,1,0,1,1, 0,0,0,1,1,0,1,1, 
0,1,0,0,1,1,0,1, 0,1,0,0,1,1,0,1, 
 
or, as read by antidiagonals: 
 
0,0,0,0,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,1,1, 
0,0,0,0,1,0,0,1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0, 
1,1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0,1,0,1,1,1,0, 
0,0,1,1,0,1,0,1,1,0,1,0,0,0,0,1,1,0,0,1,0,1,0,1,1,0,0,
 
Antidiagonal sums are simply: 
0,0,1,1,2,2,3,3,4,4,5,5,6,6,7,7,... 
  
The CD construction method may provide a more natural way to 
generate exponents of -1 that is not restricted to only 0's and 1's. 
Such a table would undoubtedly yield interesting antidiagonal sums. 
   
> There is a (IMHO beautiful!) routine for the signs,
> see http://www.jjj.de/fxt/#fxtbook
> section 20.6.1 "The Cayley-Dickson construction", p.525.
 
Agreed, it is an elegant construction. 
Also I find your paper very interesting. 
    Paul