0,1

The signs in second line of the of the table give the Thue-Morse sequence (A010060)

Joerg Arndt, fxtbook. Section "Multiplication of hypercomplex numbers" in chapter "Miscellaneous arithmetical algorithms"

Joerg Arndt, Demo program

John C. Baez, The Octonions, Bull. Amer. Math. Soc., 39 (2002), 145-205.

Multiplication table for the octonions:

Let e0,e1,...e7 be the units.

The third entry in the second row is 3+, meaning that e1*e2==+e3

The product is anti-commutative unless one factor is e0.

0 1 2 3 4 5 6 7

0: 0+ 1+ 2+ 3+ 4+ 5+ 6+ 7+

1: 1+ 0- 3- 2+ 5- 4+ 7+ 6-

2: 2+ 3+ 0- 1- 6- 7- 4+ 5+

3: 3+ 2- 1+ 0- 7- 6+ 5- 4+

4: 4+ 5+ 6+ 7+ 0- 1- 2- 3-

5: 5+ 4- 7+ 6- 1+ 0- 3+ 2-

6: 6+ 7- 4- 5+ 2+ 3- 0- 1+

7: 7+ 6+ 5- 4- 3+ 2+ 1- 0-

For the multiplication er*ec = +-ep we have p = r XOR c

The sign is given in the following array:

0 1 2 3 4 5 6 7

0: + + + + + + + +

1: + - - + - + + -

2: + + - - - - + +

3: + - + - - + - +

4: + + + + - - - -

5: + - + - + - + -

6: + - - + + - - +

7: + + - - + + - -

Now replace all + by 0 and all - by 1.

Read by antidiagonals (rising order) to obtain the sequence.

Cayley-Dickson construction:

Multiplication rule is

(a,b)*(A,b) = (a*A - B*conj(b), conj(a)*B + A*b)

where conj(a,b) := (conj(a), -b) and conj(x):=x for x real

[ Transposed rule/table is obtained if rule is changed to

(a,b)*(A,b) = (a*A - conj(B)*b, b*conj(A) + B*a) ]

// C++ (returns +1 or -1) void cp2(ulong a, ulong b, ulong &u, ulong &v) { u=a; v=b; } int CD_sign(ulong r, ulong c, ulong n) { int s = +1; start: if ( (r==0) || (c==0) ) return s; if ( c==r ) return -s; if ( c>r ) { swap2(r, c); s=-s; } n >>= 1; if ( c>=n ) cp2(c-n, r-n, r, c); else if ( r>=n ) cp2(c, r-n, r, c); goto start; } // swap2(x, y) shall swap its arguments

Cf. A096809 A010060.

Sequence in context: A079260 A025457 A093957 this_sequence A080343 A011664 A094091

Adjacent sequences: A118682 A118683 A118684 this_sequence A118686 A118687 A118688

nonn,tabl

Joerg Arndt (arndt(AT)jjj.de), May 20 2006