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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

Table of n, a(n) for n=0..104.

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: A188009 A144596 A188187 * A244063 A080343 A011664

Adjacent sequences:  A118682 A118683 A118684 * A118686 A118687 A118688

KEYWORD

nonn,tabl

AUTHOR

Joerg Arndt, May 20 2006

STATUS

approved

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Last modified May 22 05:03 EDT 2019. Contains 323473 sequences. (Running on oeis4.)