cp's OEIS Frontend

A359495 Sum of positions of 1's in binary expansion minus sum of positions of 1's in reversed binary expansion, where positions in a sequence are read starting with 1 from the left.

%I A359495 #21 Jan 09 2023 12:41:35
%S A359495 0,0,-1,0,-2,0,-2,0,-3,0,-2,1,-4,-1,-3,0,-4,0,-2,2,-4,0,-2,2,-6,-2,-4,
%T A359495 0,-6,-2,-4,0,-5,0,-2,3,-4,1,-1,4,-6,-1,-3,2,-5,0,-2,3,-8,-3,-5,0,-7,
%U A359495 -2,-4,1,-9,-4,-6,-1,-8,-3,-5,0,-6,0,-2,4,-4,2,0,6
%N A359495 Sum of positions of 1's in binary expansion minus sum of positions of 1's in reversed binary expansion, where positions in a sequence are read starting with 1 from the left.
%C A359495 Also the sum of partial sums of reversed binary expansion minus sum of partial sums of binary expansion.
%H A359495 Alois P. Heinz, <a href="/A359495/b359495.txt">Table of n, a(n) for n = 0..16383</a>
%F A359495 a(n) = A029931(n) - A230877(n).
%F A359495 If n = Sum_{i=1..k} q_i * 2^(i-1), then a(n) = Sum_{i=1..k} q_i * (2i-k-1).
%e A359495 The binary expansion of 158 is (1,0,0,1,1,1,1,0), with positions of 1's {1,4,5,6,7} with sum 23, reversed {2,3,4,5,8} with sum 22, so a(158) = 1.
%p A359495 a:= n-> (l-> add(i*(l[-i]-l[i]), i=1..nops(l)))(Bits[Split](n)):
%p A359495 seq(a(n), n=0..127);  # _Alois P. Heinz_, Jan 09 2023
%t A359495 sap[q_]:=Sum[q[[i]]*(2i-Length[q]-1),{i,Length[q]}];
%t A359495 Table[sap[IntegerDigits[n,2]],{n,0,100}]
%o A359495 (Python)
%o A359495 def A359495(n):
%o A359495     k = n.bit_length()-1
%o A359495     return sum((i<<1)-k for i, j in enumerate(bin(n)[2:]) if j=='1') # _Chai Wah Wu_, Jan 09 2023
%Y A359495 Indices of positive terms are A359401.
%Y A359495 Indices of 0's are A359402.
%Y A359495 A030190 gives binary expansion, reverse A030308.
%Y A359495 A070939 counts binary digits.
%Y A359495 A230877 adds up positions of 1's in binary expansion, reverse A029931.
%Y A359495 Cf. A000120, A048793, A053632, A222955, A231204, A291166, A326669, A326672, A326673, A359042, A359043.
%K A359495 sign,look,base
%O A359495 0,5
%A A359495 _Gus Wiseman_, Jan 05 2023