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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.

%I A118666 #26 Jan 06 2025 22:36:21
%S A118666 0,1,6,7,18,19,20,21,106,107,108,109,120,121,126,127,258,259,260,261,
%T A118666 272,273,278,279,360,361,366,367,378,379,380,381,1546,1547,1548,1549,
%U A118666 1560,1561,1566,1567,1632,1633,1638,1639,1650,1651,1652,1653,1800,1801
%N 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.
%C A118666 If p(x) is a fixed point then P(x):=(x+x^2)*p(x) and P(x)+1 are also fixed points.
%H A118666 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 1.19.3 "Fixed points of the blue code", p.52-54
%H A118666 <a href="/index/Ge#GF2X">Index entries for sequences operating on (or containing) GF(2)[X]-polynomials</a>
%e A118666 a(4)=18 corresponds to the polynomial p(x)=x^4+x (18 is 10010 in binary).
%e A118666 p(x+1) = (x+1)^4 + (x+1) = x^4 + 4*x^3 + 6*x^2 + 5*x + 2 = x^4+x = p(x);
%o A118666 (C++) /* Returns a unique fixed point for each argument: */
%o A118666 ulong A(ulong s)
%o A118666 {
%o A118666   if ( 0==s ) return 0;
%o A118666   ulong f = 1;
%o A118666   while ( s>1 ) { f ^= (f<<1); f <<= 1; f |= (s&1); s >>= 1; }
%o A118666   return f;
%o A118666 }
%o A118666 /* the elements are not produced in increasing order, but as follows:
%o A118666   0 1 6 7 20 18 21 19 120 108 126 106 121 109 127 107 272 360 ... */
%Y A118666 Cf. A193231 (the map p(x) -> p(x+1)).
%K A118666 nonn
%O A118666 0,3
%A A118666 _Joerg Arndt_, May 19 2006, May 20 2006