A231204 If n = Sum_{i=0..m} c(i)*2^i, c(i) = 0 or 1, then a(n) = Sum_{i=0..m} (m-i)*c(i).

0, 0, 0, 1, 0, 2, 1, 3, 0, 3, 2, 5, 1, 4, 3, 6, 0, 4, 3, 7, 2, 6, 5, 9, 1, 5, 4, 8, 3, 7, 6, 10, 0, 5, 4, 9, 3, 8, 7, 12, 2, 7, 6, 11, 5, 10, 9, 14, 1, 6, 5, 10, 4, 9, 8, 13, 3, 8, 7, 12, 6, 11, 10, 15, 0, 6, 5, 11, 4, 10, 9, 15, 3, 9, 8, 14, 7, 13, 12, 18, 2, 8, 7, 13, 6, 12, 11, 17, 5, 11, 10, 16, 9, 15, 14, 20, 1, 7, 6, 12

Offset

0,6

Comments

A literal interpretation of the binary numbers.

Sum of the number of digits to the left (exclusive) of each 1 in the binary expansion of n. - Gus Wiseman, Jan 09 2023

Links

Rémy Sigrist, Table of n, a(n) for n = 0..8192

Formula

a(n) = A230877(n) - A000120(n). - Gus Wiseman, Jan 09 2023

Example

For n=13 we have 1101, so we add 0+1+3=4, getting a(13)=4.

Maple

f:=proc(n) local t1, m, i;
t1:=convert(n, base, 2);
m:=nops(t1)-1;
add((m-i)*t1[i+1], i=0..m);
end; # N. J. A. Sloane, Nov 08 2013

Mathematica

Table[Total[Join@@Position[IntegerDigits[n, 2], 1]-1], {n, 0, 100}] (* Gus Wiseman, Jan 09 2023 *)

Prog

(JavaScript)
for (i=0; i<100; i++) {
s=i.toString(2);
o=0;
sl=s.length;
for (j=0; j<sl; j++) if (s.charAt(j)==1) o+=j;
document.write(o+", ");
}
(PARI) a(n) = { my (b=binary(n)); sum(k=1, #b, b[k]*(k-1)) } \\ Rémy Sigrist, Jun 25 2021
(Python)
def A230204(n): return sum(i for i, j in enumerate(bin(n)[2:]) if j=='1') # Chai Wah Wu, Jan 09 2023

Crossrefs

Cf. A029931. See A230877 for another version.

Cf. A000523.

A000120 counts 1's in binary expansion.

A023416 counts zeros in binary expansion, partial sums A059015.

A030190 gives binary expansion, reverse A030308.

A070939 counts binary digits.

A358194 counts partitions by sum of partial sums, compositions A053632.

A359359 adds up positions of zeros in binary expansion, reversed A359400.

Cf. A048793, A069010, A073642, A083652, A326669, A359043, A359402, A359495.

Sequence in context: A141198 A239621 A349444 * A180987 A092093 A197386

Adjacent sequences: A231201 A231202 A231203 * A231205 A231206 A231207

Keyword

nonn,base

Author

Jon Perry, Nov 05 2013

Extensions

Edited by N. J. A. Sloane, Nov 08 2013

Name edited by Gus Wiseman, Jan 09 2023

Status

approved