A334913 a(n) is the sum of digits of n in signed binary nonadjacent form.

0, 1, 1, 0, 1, 2, 0, 0, 1, 2, 2, -1, 0, 1, 0, 0, 1, 2, 2, 1, 2, 3, -1, -1, 0, 1, 1, -1, 0, 1, 0, 0, 1, 2, 2, 1, 2, 3, 1, 1, 2, 3, 3, -2, -1, 0, -1, -1, 0, 1, 1, 0, 1, 2, -1, -1, 0, 1, 1, -1, 0, 1, 0, 0, 1, 2, 2, 1, 2, 3, 1, 1, 2, 3, 3, 0, 1, 2, 1, 1, 2, 3, 3, 2

Offset: 0

Author

Lei Zhou, May 16 2020

Keywords

• base
• easy
• sign

Links

• Joerg Arndt, Matters Computational - Ideas, Algorithms, Source Code, 2011, Springer, pp. 61-62.
• Helmut Prodinger, On binary representations of integers with digits -1,0,1, Integers 0 (2000), #A08.

Crossrefs

Cf. A000120, A001045, A007302, A184615, A184616.

Programs

Mathematica

BBN[a_] := Module[{n = a, b}, b = IntegerDigits[n, 2]; b = Prepend[b, 0];
  l = Length[b];
  Do[If[b[[i]] == 2, b[[i]] = 0; b[[i - 1]]++,
    If[b[[i]] == 1,
     If[b[[i + 1]] == 1, b[[i - 1]]++; b[[i]] = 0;
      b[[i + 1]] = -1]]], {i, l - 1, 2, -1}];
  If[b[[1]] == 0, b = Delete[b, 1]]; b]
Table[a = BBN[i]; sod = 0; l = Length[a];
Do[sod = sod + a[[j]], {j, 1, l}]; sod, {i, 0, 83}]

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 */
}
a(n) = my(b=bin2naf(n)); return(hammingweight(b[1])-hammingweight(b[2]));
vector(99,n,a(n-1)) \\ Joerg Arndt, Jun 13 2020

Formula

a(n) = hammingweight(A184615(n)) - hammingweight(A184616(n)). - Joerg Arndt, Jun 13 2020