A079471 Fixed points of reversed binary words in reversed lexicographic order.

0, 1, 6, 10, 18, 34, 60, 66, 92, 108, 116, 130, 156, 172, 180, 204, 212, 228, 258, 284, 300, 308, 332, 340, 356, 396, 404, 420, 452, 514, 540, 556, 564, 588, 596, 612, 652, 660, 676, 708, 780, 788, 804, 836, 900, 1026, 1052, 1068, 1076, 1100, 1108, 1124

Offset

0,3

Comments

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

Links

Paolo Xausa, Table of n, a(n) for n = 0..10000

Joerg Arndt, Fxtbook, section 1.26.4 "The sequence of fixed points", p.73-74

Example

Zero is a fixed point: 0: ...........
The next few in decimal and binary form (dots for zeros), lowest (rightmost) bit has index zero are:
1: ............1
6: ..........11.
10: ........1.1.
18: .......1..1.
34: ......1...1.
60: ......1111..
66: .....1....1.
92: .....1.111..
108: ....11.11..
116: ....111.1..
130: ...1.....1.

Mathematica

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

Prog

(C++)
/* Generate the binary words lex order:
  start with zero and get successive elements via */
inline ulong prev_lexrev(ulong x)
/* Return previous word in (reversed) lex order. */
{
  ulong x0 = x & -x;
  if ( x & (x0<<1) ) x ^= x0;
  else { x0 ^= (x0<<1); x ^= x0; x |= 1; }
  return x;
}
/* To extract the fixed points, select those where
   the following function returns a nonzero value: */
ulong is_lexrev_fixed_point(ulong x)
/* Return whether x is a fixed point in the prev_lexrev() - sequence */
{
  if ( x & 1 ) { if ( 1==x ) return 1; else return 0; }
  else
  {
    ulong w = bit_count(x);
    if ( w != (w & -w) ) return 0;
    if ( 0==x ) return 1; return ( (x & -x) & w );
  }
}

Crossrefs

Sequence in context: A338122 A169873 A363788 * A134351 A307458 A338450

Adjacent sequences: A079468 A079469 A079470 * A079472 A079473 A079474

Keyword

easy,nonn

Author

Joerg Arndt, Jan 15 2003

Status

approved