A184615 Positive parts of the nonadjacent forms for n.

0, 1, 2, 4, 4, 5, 8, 8, 8, 9, 10, 16, 16, 17, 16, 16, 16, 17, 18, 20, 20, 21, 32, 32, 32, 33, 34, 32, 32, 33, 32, 32, 32, 33, 34, 36, 36, 37, 40, 40, 40, 41, 42, 64, 64, 65, 64, 64, 64, 65, 66, 68, 68, 69, 64, 64, 64, 65, 66, 64, 64, 65, 64, 64, 64, 65, 66, 68, 68, 69, 72, 72, 72, 73, 74, 80, 80, 81, 80, 80, 80, 81, 82, 84, 84, 85, 128

Offset

0,3

Comments

This sequence together with A184616 (negated negative parts) gives the (signed binary) nonadjacent form (NAF) of n, see fxtbook link.

No two adjacent bits in the binary representations of a(n) are 1.

No two adjacent bits in the binary representations of a(n)+A184616(n) are 1.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..8192

Pages 61-62 of Matters Computational (The Fxtbook).

Formula

a(n) - A184616(n) = n

a(n) + A184616(n) = A184617(n)

Example

The first few nonadjacent forms (NAF) are
(dots are used for zeros for better readability):
     n     binary(n)  NAF(n)
   0:    .......    .......      0 =
   1:    ......1    ......P      1 =  +1
   2:    .....1.    .....P.      2 =  +2
   3:    .....11    ....P.M      3 =  +4 -1
   4:    ....1..    ....P..      4 =  +4
   5:    ....1.1    ....P.P      5 =  +4 +1
   6:    ....11.    ...P.M.      6 =  +8 -2
   7:    ....111    ...P..M      7 =  +8 -1
   8:    ...1...    ...P...      8 =  +8
   9:    ...1..1    ...P..P      9 =  +8 +1
  10:    ...1.1.    ...P.P.     10 =  +8 +2
  11:    ...1.11    ..P.M.M     11 =  +16 -4 -1
  12:    ...11..    ..P.M..     12 =  +16 -4
  13:    ...11.1    ..P.M.P     13 =  +16 -4 +1
  14:    ...111.    ..P..M.     14 =  +16 -2
  15:    ...1111    ..P...M     15 =  +16 -1
  16:    ..1....    ..P....     16 =  +16
  17:    ..1...1    ..P...P     17 =  +16 +1
  18:    ..1..1.    ..P..P.     18 =  +16 +2
This sequence gives the words obtained by keeping the 'P' (sum of positive terms in rightmost column), keeping the 'M' gives A184616 (negative sum of negative terms in rightmost column).

Mathematica

bin2naf[x_] := Module[{xh, x3, c, np, nm},
  xh = BitShiftRight[x, 1];
  x3 = x + xh;
  c = BitXor[xh, x3];
  np = BitAnd[x3, c];
  nm = BitAnd[xh, c];
  Return[{np, nm}]];
a[n_] := bin2naf[n][[1]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, May 30 2019, from PARI *)

Prog

(PARI)
bin2naf(x)=
{ /* Compute (nonadjacent) signed binary representation of x: */
    local(xh, x3, c, np, nm);
    xh = x >> 1;
    x3 = x + xh;
    c = bitxor(xh, x3);
    np = bitand(x3, c);  /* bits == +1 */
    nm = bitand(xh, c);  /* bits == -1 */
    return([np, nm]);  /* np-nm==x */
}
{ for(n=0, 100, v = bin2naf(n); print1(v[1], ", "); ); } /* show terms */
{ for(n=0, 100, v = bin2naf(n); print1(v[2], ", "); ); } /* terms of A184616 */
{ for(n=0, 100, v = bin2naf(n); print1(v[1]+v[2], ", "); ); } /* terms of A184617 */
{ for(n=0, 100, v = bin2naf(n); print1(v[1]-v[2], ", "); ); }  /* == n */

Crossrefs

A184616 (negated negative parts), A184617 (sums of both parts =A184615+A184616).

A007302 gives the number of nonzero bits ('M' and 'P' in example).

Sequence in context: A327625 A084824 A344710 * A151969 A261393 A366541

Adjacent sequences: A184612 A184613 A184614 * A184616 A184617 A184618

Keyword

nonn

Author

Joerg Arndt, Jan 18 2011

Status

approved