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

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Index entries for sequences related to binary expansion of n

Formula

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

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

Example

+---+-----+---+---+---+---+----------+
| 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.

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]]

Prog

(Python)
def A301896(n): return (2+(n+1)*(m:=(n+1).bit_length())-(1<<m)-(k:=sum(i.bit_count() for i in range(1, n+1))))*k # Chai Wah Wu, Mar 01 2023

Crossrefs

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

Sequence in context: A034733 A365964 A152233 * A053303 A097164 A133628

Adjacent sequences: A301893 A301894 A301895 * A301897 A301898 A301899

Keyword

nonn,base

Author

Ilya Gutkovskiy, Mar 28 2018

Status

approved