A344985 Sum of imbalance of all initial subsequences of the binary representation of n.

1, 1, 3, 2, 2, 4, 6, 4, 2, 2, 4, 4, 6, 8, 10, 7, 5, 3, 3, 3, 3, 5, 7, 5, 5, 7, 9, 9, 11, 13, 15, 11, 9, 7, 5, 5, 3, 3, 5, 5, 3, 3, 5, 5, 7, 9, 11, 7, 5, 5, 7, 7, 9, 11, 13, 9, 11, 13, 15, 15, 17, 19, 21, 16, 14, 12, 10, 10, 8, 6, 6, 8, 6, 4, 4, 4, 4, 6, 8, 8

Offset: 1

Author

André Engels, Jun 04 2021

Keywords

• nonn
• base
• look
• easy

Comments

The imbalance of a sequence of zeros and ones is given as the absolute value of the difference between the number of zeros and the number of ones (that is, the absolute value of A037861 if n is the sequence A007088).

Examples

			9 in binary is 1001. The imbalance of 1 is 1, the imbalance of 10 is 0, the imbalance of 100 is 1 and the imbalance of 1001 is 0. thus a(9)=1+0+1+0=2.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..16384

Crossrefs

Cf. A007088, A037861.

Programs

Maple

imbal := proc(L)
     abs(numboccur(0, L) - numboccur(1, L))
end proc:
A344985 := proc(n)
    dgs := ListTools[Reverse](convert(n,base,2)) ;
    add(imbal( dgs[1..l]),l=1..nops(dgs)) ;
end proc: # R. J. Mathar, Jul 07 2021
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 0, a(iquo(n, 2))+
      (l-> abs(add(1-2*i, i=l)))(Bits[Split](n)))
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Jul 07 2021

PARI

a(n) = my(b=binary(n)); sum(k=1, #b, abs(sum(j=1, k, b[j]==1) - sum(j=1, k, b[j]==0))); \\ Michel Marcus, Jun 09 2021

Python

def A344985(n):
    s, c, b = bin(n)[2:], 0, 0
    for x in s:
        b += 1 if x == '1' else -1
        c += abs(b)
    return c # Chai Wah Wu, Jul 07 2021

Formula

a(n) = a(floor(n/2)) + |A037861(n)|. - R. J. Mathar, Jul 07 2021