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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). 19
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 (list; graph; refs; listen; history; text; internal format)
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
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.
Sequence in context: A141198 A239621 A349444 * A180987 A092093 A197386
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

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Last modified February 22 06:21 EST 2024. Contains 370240 sequences. (Running on oeis4.)