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

0,6

Comments

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

Links

Table of n, a(n) for n=0..83.

Formula

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

Example

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

Mathematica

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

Prog

(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

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.

Sequence in context: A319571 A086966 A140080 * A065359 A087372 A036431

Adjacent sequences: A345924 A345925 A345926 * A345928 A345929 A345930

Keyword

sign

Author

Gus Wiseman, Jul 14 2021

Status

approved