A079471 - OEIS

%I #18 Feb 18 2024 01:26:01

%S 0,1,6,10,18,34,60,66,92,108,116,130,156,172,180,204,212,228,258,284,

%T 300,308,332,340,356,396,404,420,452,514,540,556,564,588,596,612,652,

%U 660,676,708,780,788,804,836,900,1026,1052,1068,1076,1100,1108,1124

%N Fixed points of reversed binary words in reversed lexicographic order.

%C These are 0 and the words where the bit count is 2^i where i is the index of the lowest set bit.

%H Paolo Xausa, <a href="/A079471/b079471.txt">Table of n, a(n) for n = 0..10000</a>

%H Joerg Arndt, <a href="http://www.jjj.de/fxt/fxtpage.html#fxtbook">Fxtbook</a>, section 1.26.4 "The sequence of fixed points", p.73-74

%e Zero is a fixed point: 0: ...........

%e The next few in decimal and binary form (dots for zeros), lowest (rightmost) bit has index zero are:

%e 1: ............1

%e 6: ..........11.

%e 10: ........1.1.

%e 18: .......1..1.

%e 34: ......1...1.

%e 60: ......1111..

%e 66: .....1....1.

%e 92: .....1.111..

%e 108: ....11.11..

%e 116: ....111.1..

%e 130: ...1.....1.

%t Join[{0},Select[Range[10^4],DigitCount[#,2,1]==2^IntegerExponent[#,2]&]] (* _Paolo Xausa_, Nov 13 2023 *)

%o (C++)

%o /* Generate the binary words lex order:

%o   start with zero and get successive elements via */

%o inline ulong prev_lexrev(ulong x)

%o /* Return previous word in (reversed) lex order. */

%o {

%o   ulong x0 = x & -x;

%o   if ( x & (x0<<1) ) x ^= x0;

%o   else { x0 ^= (x0<<1); x ^= x0; x |= 1; }

%o   return x;

%o }

%o /* To extract the fixed points, select those where

%o    the following function returns a nonzero value: */

%o ulong is_lexrev_fixed_point(ulong x)

%o /* Return whether x is a fixed point in the prev_lexrev() - sequence */

%o {

%o   if ( x & 1 ) { if ( 1==x ) return 1; else return 0; }

%o   else

%o   {

%o     ulong w = bit_count(x);

%o     if ( w != (w & -w) ) return 0;

%o     if ( 0==x ) return 1; return ( (x & -x) & w );

%o   }

%o }

%K easy,nonn

%O 0,3

%A _Joerg Arndt_, Jan 15 2003