A345927 Alternating sum of the binary expansion of n (row n of A030190). Replace 2^k with (-1)^(A070939(n)-k) in the binary expansion of n (compare to the definition of A065359).

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

Offset: 0

Author

Gus Wiseman, Jul 14 2021

Keywords

• sign

Comments

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

Examples

			The binary expansion of 53 is (1,1,0,1,0,1), so a(53) = 1 - 1 + 0 - 1 + 0 - 1 = -2.
		

Crossrefs

Binary expansions of each nonnegative integer are the rows of A030190.

The positions of 0's are A039004.

The version for prime factors is A071321 (reverse: A071322).

Positions of first appearances are A086893.

The version for standard compositions is A124754 (reverse: A344618).

The version for prime multiplicities is A316523.

The version for prime indices is A316524 (reverse: A344616).

A003714 lists numbers with no successive binary indices.

A070939 gives the length of an integer's binary expansion.

A103919 counts partitions by sum and alternating sum.

A328594 lists numbers whose binary expansion is aperiodic.

A328595 lists numbers whose reversed binary expansion is a necklace.

Cf. A000037, A000070, A000120, A027187, A028260, A065359, A069010, A116406, A121016, A191232, A245563, A344609, A344619.

Programs

Mathematica

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[ats[IntegerDigits[n,2]],{n,0,100}]

PARI

a(n) = subst(Pol(Vecrev(binary(n))), x, -1); \\ Michel Marcus, Jul 19 2021

Python

def a(n): return sum((-1)**k for k, bi in enumerate(bin(n)[2:]) if bi=='1')
print([a(n) for n in range(84)]) # Michael S. Branicky, Jul 19 2021

Formula

a(n) = (-1)^(A070939(n)-1)*A065359(n).