A118666 Binary polynomials p(x) that are fixed points of the map p(x) -> p(x+1), evaluated as polynomials over Z at x=2.

0, 1, 6, 7, 18, 19, 20, 21, 106, 107, 108, 109, 120, 121, 126, 127, 258, 259, 260, 261, 272, 273, 278, 279, 360, 361, 366, 367, 378, 379, 380, 381, 1546, 1547, 1548, 1549, 1560, 1561, 1566, 1567, 1632, 1633, 1638, 1639, 1650, 1651, 1652, 1653, 1800, 1801

Offset

0,3

Comments

If p(x) is a fixed point then P(x):=(x+x^2)*p(x) and P(x)+1 are also fixed points.

Links

Table of n, a(n) for n=0..49.

Joerg Arndt, Matters Computational (The Fxtbook), section 1.19.3 "Fixed points of the blue code", p.52-54

Index entries for sequences operating on (or containing) GF(2)[X]-polynomials

Example

a(4)=18 corresponds to the polynomial p(x)=x^4+x (18 is 10010 in binary).
p(x+1) = (x+1)^4 + (x+1) = x^4 + 4*x^3 + 6*x^2 + 5*x + 2 = x^4+x = p(x);

Prog

/* C++ function that returns a unique fixed point for each argument: */
ulong A(ulong s)
{
  if ( 0==s ) return 0;
  ulong f = 1;
  while ( s>1 ) { f ^= (f<<1); f <<= 1; f |= (s&1); s >>= 1; }
  return f;
}
/* the elements are not produced in increasing order, but as follows:
  0 1 6 7 20 18 21 19 120 108 126 106 121 109 127 107 272 360 ... */

Crossrefs

Cf. A193231 (the map p(x) -> p(x+1)).

Sequence in context: A219853 A250196 A260563 * A315847 A030746 A315848

Adjacent sequences: A118663 A118664 A118665 * A118667 A118668 A118669

Keyword

nonn

Author

Joerg Arndt, May 19 2006, May 20 2006

Status

approved