%I #14 Sep 03 2020 15:25:14

%S -1,0,0,2,1,2,3,6,4,4,4,6,6,8,10,14,11,10,9,10,9,10,11,14,13,14,15,18,

%T 19,22,25,30,26,24,22,22,20,20,20,22,20,20,20,22,22,24,26,30,28,28,28,

%U 30,30,32,34,38,38,40,42,46,48,52,56,62,57,54,51,50,47,46,45,46,43,42,41,42

%N a(n) = total number of 1's minus total number of 0's in binary expansions of 0, ..., n.

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%F G.f.: -1/(1 - x) + (1/(1 - x)^2)*Sum_{k>=0} x^(2^k)*(1 - x^(2^k))/(1 + x^(2^k)).

%F a(n) = A000788(n) - A059015(n).

%F a(n) = A268289(n) - 1.

%F a(A000079(n)) = A000295(n).

%e +---+-----+---+---+---+---+------------+

%e | n | bin.|1's|sum|0's|sum| a(n) |

%e +---+-----+---+---+---+---+------------+

%e | 0 | 0 | 0 | 0 | 1 | 1 | 0 - 1 =-1 |

%e | 1 | 1 | 1 | 1 | 0 | 1 | 1 - 1 = 0 |

%e | 2 | 10 | 1 | 2 | 1 | 2 | 2 - 2 = 0 |

%e | 3 | 11 | 2 | 4 | 0 | 2 | 4 - 2 = 2 |

%e | 4 | 100 | 1 | 5 | 2 | 4 | 5 - 4 = 1 |

%e | 5 | 101 | 2 | 7 | 1 | 5 | 7 - 5 = 2 |

%e | 6 | 110 | 2 | 9 | 1 | 6 | 9 - 6 = 3 |

%e +---+-----+---+---+---+---+------------+

%e bin. - n written in base 2;

%e 1's - number of 1's in binary expansion of n;

%e 0's - number of 0's in binary expansion of n;

%e sum - total number of 1's (or 0's) in binary expansions of 0, ..., n.

%t Accumulate[DigitCount[Range[0, 75], 2, 1] - DigitCount[Range[0, 75], 2, 0]]

%o (Python)

%o def A301336(n):

%o return sum(2*bin(i).count('1')-len(bin(i))+2 for i in range(n+1)) # _Chai Wah Wu_, Sep 03 2020

%Y Cf. A000079, A000120, A000295, A000788, A001855, A023416, A037861, A059015, A083652, A145037, A268289, A301896.

%K sign,base

%O 0,4

%A _Ilya Gutkovskiy_, Mar 28 2018