%I #32 Jan 11 2023 08:46:57
%S 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,
%T 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,
%U 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
%N 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).
%C A literal interpretation of the binary numbers.
%C Sum of the number of digits to the left (exclusive) of each 1 in the binary expansion of n. - _Gus Wiseman_, Jan 09 2023
%H Rémy Sigrist, <a href="/A231204/b231204.txt">Table of n, a(n) for n = 0..8192</a>
%F a(n) = A230877(n) - A000120(n). - _Gus Wiseman_, Jan 09 2023
%e For n=13 we have 1101, so we add 0+1+3=4, getting a(13)=4.
%p f:=proc(n) local t1,m,i;
%p t1:=convert(n,base,2);
%p m:=nops(t1)-1;
%p add((m-i)*t1[i+1], i=0..m);
%p end; # _N. J. A. Sloane_, Nov 08 2013
%t Table[Total[Join@@Position[IntegerDigits[n,2],1]-1],{n,0,100}] (* _Gus Wiseman_, Jan 09 2023 *)
%o (JavaScript)
%o for (i=0;i<100;i++) {
%o s=i.toString(2);
%o o=0;
%o sl=s.length;
%o for (j=0;j<sl;j++) if (s.charAt(j)==1) o+=j;
%o document.write(o+", ");
%o }
%o (PARI) a(n) = { my (b=binary(n)); sum(k=1, #b, b[k]*(k-1)) } \\ _Rémy Sigrist_, Jun 25 2021
%o (Python)
%o def A230204(n): return sum(i for i, j in enumerate(bin(n)[2:]) if j=='1') # _Chai Wah Wu_, Jan 09 2023
%Y Cf. A029931. See A230877 for another version.
%Y Cf. A000523.
%Y A000120 counts 1's in binary expansion.
%Y A023416 counts zeros in binary expansion, partial sums A059015.
%Y A030190 gives binary expansion, reverse A030308.
%Y A070939 counts binary digits.
%Y A358194 counts partitions by sum of partial sums, compositions A053632.
%Y A359359 adds up positions of zeros in binary expansion, reversed A359400.
%Y Cf. A048793, A069010, A073642, A083652, A326669, A359043, A359402, A359495.
%K nonn,base
%O 0,6
%A _Jon Perry_, Nov 05 2013
%E Edited by _N. J. A. Sloane_, Nov 08 2013
%E Name edited by _Gus Wiseman_, Jan 09 2023