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%I #22 May 22 2023 06:22:13
%S 0,0,0,0,1,0,0,0,1,0,0,1,1,0,0,0,0,0,1,1,0,0,1,0,1,1,0,0,0,1,0,0,1,1,
%T 0,0,0,0,1,1,1,0,0,1,0,0,0,1,0,0,1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0,0,1,
%U 0,0,0,1,0,1,1,1,0,0,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,0,1,0,1,1,1,0,0,0
%N Signs of entries in the multiplication table for hypercomplex numbers with Cayley-Dickson construction (by antidiagonals).
%C The signs in the second line of the table give the Thue-Morse sequence (A010060).
%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 39.14.1 "The Cayley-Dickson construction", pp.815-818
%H Joerg Arndt, <a href="http://www.jjj.de/fxt/demo/arith/#cayley-dickson">Demo program</a>
%H John C. Baez, <a href="http://math.ucr.edu/home/baez/octonions/">The Octonions</a>, Bull. Amer. Math. Soc., 39 (2002), 145-205.
%e Multiplication table for the octonions:
%e Let e0,e1,...e7 be the units.
%e The third entry in the second row is 3+, meaning that e1*e2==+e3.
%e The product is anti-commutative unless one factor is e0.
%e 0 1 2 3 4 5 6 7
%e 0: 0+ 1+ 2+ 3+ 4+ 5+ 6+ 7+
%e 1: 1+ 0- 3- 2+ 5- 4+ 7+ 6-
%e 2: 2+ 3+ 0- 1- 6- 7- 4+ 5+
%e 3: 3+ 2- 1+ 0- 7- 6+ 5- 4+
%e 4: 4+ 5+ 6+ 7+ 0- 1- 2- 3-
%e 5: 5+ 4- 7+ 6- 1+ 0- 3+ 2-
%e 6: 6+ 7- 4- 5+ 2+ 3- 0- 1+
%e 7: 7+ 6+ 5- 4- 3+ 2+ 1- 0-
%e For the multiplication er*ec = +-ep we have p = r XOR c
%e The sign is given in the following array:
%e 0 1 2 3 4 5 6 7
%e 0: + + + + + + + +
%e 1: + - - + - + + -
%e 2: + + - - - - + +
%e 3: + - + - - + - +
%e 4: + + + + - - - -
%e 5: + - + - + - + -
%e 6: + - - + + - - +
%e 7: + + - - + + - -
%e Now replace all + by 0 and all - by 1.
%e Read by antidiagonals (rising order) to obtain the sequence.
%e Cayley-Dickson construction:
%e Multiplication rule is
%e (a,b)*(A,b) = (a*A - B*conj(b), conj(a)*B + A*b)
%e where conj(a,b) := (conj(a), -b) and conj(x):=x for x real
%e [ Transposed rule/table is obtained if rule is changed to
%e (a,b)*(A,b) = (a*A - conj(B)*b, b*conj(A) + B*a) ]
%o (C++)
%o void cp2(ulong a, ulong b, ulong &u, ulong &v) { u=a; v=b; }
%o int CD_sign(ulong r, ulong c, ulong n) // (returns +1 or -1)
%o {
%o int s = +1;
%o while ( true )
%o {
%o if ( (r==0) || (c==0) ) return s;
%o if ( c==r ) return -s;
%o if ( c>r ) { std::swap(r,c); s=-s; }
%o n >>= 1;
%o if ( c>=n ) cp2(c-n, r-n, r, c);
%o else if ( r>=n ) cp2(c, r-n, r, c);
%o }
%o }
%Y Cf. A096809, A010060.
%K nonn,tabl
%O 0,1
%A _Joerg Arndt_, May 20 2006
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