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 A118685 Signs of entries in the multiplication table for hypercomplex numbers with Cayley-Dickson construction (by antidiagonals). 0
 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, 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, 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS The signs in the second line of the table give the Thue-Morse sequence (A010060). LINKS Joerg Arndt, Matters Computational (The Fxtbook), section 39.14.1 "The Cayley-Dickson construction", pp.815-818 Joerg Arndt, Demo program John C. Baez, The Octonions, Bull. Amer. Math. Soc., 39 (2002), 145-205. EXAMPLE 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) ] PROG /* C++ (returns +1 or -1) */ void cp2(ulong a, ulong b, ulong &u, ulong &v) { u=a; v=b; } /* auxiliary func. */ int CD_sign(ulong r, ulong c, ulong n) {   int s = +1;   while ( true )   {       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);   } } /* Note: the function void swap2(ulong &x, ulong &y) shall swap its arguments */ CROSSREFS Cf. A096809, A010060. Sequence in context: A093957 A144601 A144596 * A080343 A011664 A179831 Adjacent sequences:  A118682 A118683 A118684 * A118686 A118687 A118688 KEYWORD nonn,tabl AUTHOR Joerg Arndt, May 20 2006 STATUS approved

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