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A231204
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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).
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19
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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
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OFFSET
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0,6
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COMMENTS
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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
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LINKS
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FORMULA
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EXAMPLE
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For n=13 we have 1101, so we add 0+1+3=4, getting a(13)=4.
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MAPLE
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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);
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MATHEMATICA
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Table[Total[Join@@Position[IntegerDigits[n, 2], 1]-1], {n, 0, 100}] (* Gus Wiseman, Jan 09 2023 *)
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PROG
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(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
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CROSSREFS
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A000120 counts 1's in binary expansion.
A358194 counts partitions by sum of partial sums, compositions A053632.
A359359 adds up positions of zeros in binary expansion, reversed A359400.
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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