A301896 a(n) = product of total number of 0's and total number of 1's in binary expansions of 0, ..., n.

0, 1, 4, 8, 20, 35, 54, 72, 117, 165, 221, 280, 352, 425, 504, 576, 726, 875, 1036, 1200, 1386, 1575, 1776, 1976, 2214, 2451, 2700, 2944, 3216, 3479, 3750, 4000, 4455, 4897, 5355, 5808, 6300, 6789, 7296, 7800, 8364, 8925, 9504, 10080, 10695, 11305, 11931, 12544, 13260, 13965, 14688

Offset: 0

Author

Ilya Gutkovskiy, Mar 28 2018

Keywords

• nonn
• base

Examples

			+---+-----+---+---+---+---+----------+
| n | bin.|0's|sum|1's|sum|   a(n)   |
+---+-----+---+---+---+---+----------+
| 0 |   0 | 1 | 1 | 0 | 0 | 1*0 =  0 |
| 1 |   1 | 0 | 1 | 1 | 1 | 1*1 =  1 |
| 2 |  10 | 1 | 2 | 1 | 2 | 2*2 =  4 |
| 3 |  11 | 0 | 2 | 2 | 4 | 2*4 =  8 |
| 4 | 100 | 2 | 4 | 1 | 5 | 4*5 = 20 |
| 5 | 101 | 1 | 5 | 2 | 7 | 5*7 = 35 |
| 6 | 110 | 1 | 6 | 2 | 9 | 6*9 = 54 |
+---+-----+---+---+---+---+----------+
bin. - n written in base 2;
0's - number of 0's in binary expansion of n;
1's - number of 1's in binary expansion of n;
sum - total number of 0's (or 1's) in binary expansions of 0, ..., n.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A000120, A000788, A023416, A059015, A071295, A301336.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, [1, 0], b(n-1)+
     (l-> [add(1-i, i=l), add(i, i=l)])(Bits[Split](n)))
    end:
a:= n-> (l-> l[1]*l[2])(b(n)):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 01 2023

Mathematica

Accumulate[DigitCount[Range[0, 50], 2, 0]] Accumulate[DigitCount[Range[0, 50], 2, 1]]

Python

def A301896(n): return (2+(n+1)*(m:=(n+1).bit_length())-(1<Chai Wah Wu, Mar 01 2023

Python

def A301896(n): return (a:=(n+1)*n.bit_count()+(sum((m:=1<>j)-(r if n<<1>=m*(r:=k<<1|1) else 0)) for j in range(1,n.bit_length()+1))>>1))*(2+(n+1)*(t:=(n+1).bit_length())-(1<Chai Wah Wu, Nov 11 2024

Formula

a(n) = A059015(n)*A000788(n).

a(2^k-1) = 2^(k-2)*(2^k*(k - 2) + 4)*k.