cp's OEIS Frontend

A301336 a(n) = total number of 1's minus total number of 0's in binary expansions of 0, ..., n.

%I A301336 #18 Nov 11 2024 15:43:19
%S A301336 -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 A301336 19,22,25,30,26,24,22,22,20,20,20,22,20,20,20,22,22,24,26,30,28,28,28,
%U A301336 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 A301336 a(n) = total number of 1's minus total number of 0's in binary expansions of 0, ..., n.
%H A301336 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A301336 G.f.: -1/(1 - x) + (1/(1 - x)^2)*Sum_{k>=0} x^(2^k)*(1 - x^(2^k))/(1 + x^(2^k)).
%F A301336 a(n) = A000788(n) - A059015(n).
%F A301336 a(n) = A268289(n) - 1.
%F A301336 a(A000079(n)) = A000295(n).
%e A301336 +---+-----+---+---+---+---+------------+
%e A301336 | n | bin.|1's|sum|0's|sum|    a(n)    |
%e A301336 +---+-----+---+---+---+---+------------+
%e A301336 | 0 |   0 | 0 | 0 | 1 | 1 | 0 - 1 =-1  |
%e A301336 | 1 |   1 | 1 | 1 | 0 | 1 | 1 - 1 = 0  |
%e A301336 | 2 |  10 | 1 | 2 | 1 | 2 | 2 - 2 = 0  |
%e A301336 | 3 |  11 | 2 | 4 | 0 | 2 | 4 - 2 = 2  |
%e A301336 | 4 | 100 | 1 | 5 | 2 | 4 | 5 - 4 = 1  |
%e A301336 | 5 | 101 | 2 | 7 | 1 | 5 | 7 - 5 = 2  |
%e A301336 | 6 | 110 | 2 | 9 | 1 | 6 | 9 - 6 = 3  |
%e A301336 +---+-----+---+---+---+---+------------+
%e A301336 bin. - n written in base 2;
%e A301336 1's - number of 1's in binary expansion of n;
%e A301336 0's - number of 0's in binary expansion of n;
%e A301336 sum - total number of 1's (or 0's) in binary expansions of 0, ..., n.
%p A301336 a:= proc(n) option remember; `if`(n=0, -1,
%p A301336       a(n-1)+add(2*i-1, i=Bits[Split](n)))
%p A301336     end:
%p A301336 seq(a(n), n=0..75);  # _Alois P. Heinz_, Nov 11 2024
%t A301336 Accumulate[DigitCount[Range[0, 75], 2, 1] - DigitCount[Range[0, 75], 2, 0]]
%o A301336 (Python)
%o A301336 def A301336(n):
%o A301336     return sum(2*bin(i).count('1')-len(bin(i))+2 for i in range(n+1)) # _Chai Wah Wu_, Sep 03 2020
%o A301336 (Python)
%o A301336 def A301336(n): return (n+1)*((n.bit_count()<<1)-(t:=(n+1).bit_length()))+(1<<t)+sum((m:=1<<j)*((k:=n>>j)-(r if n<<1>=m*(r:=k<<1|1) else 0)) for j in range(1,n.bit_length()+1))-2 # _Chai Wah Wu_, Nov 11 2024
%Y A301336 Cf. A000079, A000120, A000295, A000788, A001855, A023416, A037861, A059015, A083652, A145037, A268289, A301896.
%K A301336 sign,base
%O A301336 0,4
%A A301336 _Ilya Gutkovskiy_, Mar 28 2018