A145037 - OEIS

%I #81 May 14 2024 06:10:49

%S 0,1,0,2,-1,1,1,3,-2,0,0,2,0,2,2,4,-3,-1,-1,1,-1,1,1,3,-1,1,1,3,1,3,3,

%T 5,-4,-2,-2,0,-2,0,0,2,-2,0,0,2,0,2,2,4,-2,0,0,2,0,2,2,4,0,2,2,4,2,4,

%U 4,6,-5,-3,-3,-1,-3,-1,-1,1,-3,-1,-1,1,-1,1,1,3,-3,-1,-1,1,-1,1,1,3,-1,1,1

%N Number of 1's minus number of 0's in the binary representation of n.

%C Column 2 of A144912 (which begins at n = 2).

%C Zeros in that column correspond to A031443.

%H <a href="/A145037/b145037.txt">Table of n, a(n) for n = 0..2^14</a>

%F a(n) = -A037861(n) for n >= 1.

%F a(n) = Sum_{i=1..k} (2*b[i] - 1) where b is the binary expansion of n and k is the number of bits in this binary expansion. - _Michel Marcus_, Jun 28 2021

%F From _Aayush Soni_ Feb 12 2022: (Start)

%F Upper bound: a(n) <= floor(log_2(n+1)).

%F Lower bound: For n > 0, a(n) >= 1 - floor(log_2(n)).

%F If n is even, a(2^n) to a(2^(n+1)-1) inclusive are all odd and vice versa. (End)

%e From _Michel Marcus_, Feb 12 2022: (Start)

%e Viewed as an irregular triangle:

%e    0;

%e    1;

%e    0,  2;

%e   -1,  1,  1, 3;

%e   -2,  0,  0, 2,  0, 2, 2, 4;

%e   -3, -1, -1, 1, -1, 1, 1, 3, -1, 1, 1, 3, 1, 3, 3, 5;

%e   ... (End)

%p a:= n-> add(2*i-1, i=Bits[Split](n)):

%p seq(a(n), n=0..90);  # _Alois P. Heinz_, Jan 18 2022

%t Join[{0}, Table[Count[#, 1] - Count[#, 0] &[IntegerDigits[n, 2]], {n, 1, 90}]] (* _Robert P. P. McKone_, Feb 12 2022 *)

%o (Haskell)

%o a145037 0 = 0

%o a145037 n = a145037 n' + 2*m - 1 where (n', m) = divMod n 2

%o -- _Reinhard Zumkeller_, Jun 16 2011

%o (PARI) A145037(n)=hammingweight(n)*2-logint(n<<1+!n,2) \\ _M. F. Hasler_, Mar 08 2018

%o (Python)

%o result = [0]

%o for n in range (1, 2**14 + 1):

%o     result.append(bin(n)[2:].count("1") - bin(n)[2:].count("0"))

%o print(result[0:129]) # _Karl-Heinz Hofmann_, Jan 18 2022

%o (Python)

%o def a(n): return (n.bit_count()<<1) - n.bit_length()

%o print([a(n) for n in range(1, 2**14+1)]) # _Michael S. Branicky_, May 14 2024

%o (C#)

%o int A145037(int n)  {

%o     int result = 0;

%o     while(n > 0)  {

%o         result += 2 * (n % 2) - 1;

%o         n /= 2;

%o     }

%o     return result;

%o } \\ _Frank Hollstein_, Dec 08 2022

%Y Cf. A037861 (negated), A031443 (indices of 0's), A144912, A000120.

%Y Cf. A269735 (first differences), A268289 (partial sums).

%Y Column k=1 of A360099.

%K sign,base,easy

%O 0,4

%A _Reikku Kulon_, Sep 30 2008

%E Renamed (using a Mar 08 2018 comment from _M. F. Hasler_) and edited by _Jon E. Schoenfield_, Jun 29 2021