cp's OEIS Frontend

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).

%I A345927 #17 Jul 20 2021 03:30:45
%S A345927 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,
%T A345927 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,
%U A345927 1,2,0,1,2,3,1,2,0,1,-1,0,1,2,0,1,2,3,1,2
%N 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).
%C A345927 The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
%F A345927 a(n) = (-1)^(A070939(n)-1)*A065359(n).
%e A345927 The binary expansion of 53 is (1,1,0,1,0,1), so a(53) = 1 - 1 + 0 - 1 + 0 - 1 = -2.
%t A345927 ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t A345927 Table[ats[IntegerDigits[n,2]],{n,0,100}]
%o A345927 (PARI) a(n) = subst(Pol(Vecrev(binary(n))), x, -1); \\ _Michel Marcus_, Jul 19 2021
%o A345927 (Python)
%o A345927 def a(n): return sum((-1)**k for k, bi in enumerate(bin(n)[2:]) if bi=='1')
%o A345927 print([a(n) for n in range(84)]) # _Michael S. Branicky_, Jul 19 2021
%Y A345927 Binary expansions of each nonnegative integer are the rows of A030190.
%Y A345927 The positions of 0's are A039004.
%Y A345927 The version for prime factors is A071321 (reverse: A071322).
%Y A345927 Positions of first appearances are A086893.
%Y A345927 The version for standard compositions is A124754 (reverse: A344618).
%Y A345927 The version for prime multiplicities is A316523.
%Y A345927 The version for prime indices is A316524 (reverse: A344616).
%Y A345927 A003714 lists numbers with no successive binary indices.
%Y A345927 A070939 gives the length of an integer's binary expansion.
%Y A345927 A103919 counts partitions by sum and alternating sum.
%Y A345927 A328594 lists numbers whose binary expansion is aperiodic.
%Y A345927 A328595 lists numbers whose reversed binary expansion is a necklace.
%Y A345927 Cf. A000037, A000070, A000120, A027187, A028260, A065359, A069010, A116406, A121016, A191232, A245563, A344609, A344619.
%K A345927 sign
%O A345927 0,6
%A A345927 _Gus Wiseman_, Jul 14 2021