A334913 - OEIS

%I #27 Jul 16 2021 01:34:57

%S 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,

%T 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,

%U 0,1,2,2,1,2,3,1,1,2,3,3,0,1,2,1,1,2,3,3,2

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

%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational - Ideas, Algorithms, Source Code</a>, 2011, Springer, pp. 61-62.

%H Helmut Prodinger, <a href="http://www.emis.de/journals/INTEGERS/papers/a8/a8.Abstract.html">On binary representations of integers with digits -1,0,1</a>, Integers 0 (2000), #A08.

%F a(n) = hammingweight(A184615(n)) - hammingweight(A184616(n)). - _Joerg Arndt_, Jun 13 2020

%t BBN[a_] := Module[{n = a, b}, b = IntegerDigits[n, 2]; b = Prepend[b, 0];

%t   l = Length[b];

%t   Do[If[b[[i]] == 2, b[[i]] = 0; b[[i - 1]]++,

%t     If[b[[i]] == 1,

%t      If[b[[i + 1]] == 1, b[[i - 1]]++; b[[i]] = 0;

%t       b[[i + 1]] = -1]]], {i, l - 1, 2, -1}];

%t   If[b[[1]] == 0, b = Delete[b, 1]]; b]

%t Table[a = BBN[i]; sod = 0; l = Length[a];

%t Do[sod = sod + a[[j]], {j, 1, l}]; sod, {i, 0, 83}]

%o (PARI)

%o bin2naf(x)=

%o { /* Compute (nonadjacent) signed binary representation of x: */

%o     local(xh, x3, c, np, nm);

%o     xh = x >> 1;

%o     x3 = x + xh;

%o     c = bitxor(xh, x3);

%o     np = bitand(x3, c);  /* bits == +1 */

%o     nm = bitand(xh, c);  /* bits == -1 */

%o     return([np, nm]);  /* np-nm==x */

%o }

%o a(n) = my(b=bin2naf(n)); return(hammingweight(b[1])-hammingweight(b[2]));

%o vector(99,n,a(n-1)) \\ _Joerg Arndt_, Jun 13 2020

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

%K base,easy,sign

%O 0,6

%A _Lei Zhou_, May 16 2020