cp's OEIS Frontend

A359402 Numbers whose binary expansion and reversed binary expansion have the same sum of positions of 1's, where positions in a sequence are read starting with 1 from the left.

%I A359402 #12 Jan 08 2023 01:16:26
%S A359402 0,1,3,5,7,9,15,17,21,27,31,33,45,51,63,65,70,73,78,85,93,99,107,119,
%T A359402 127,129,150,153,165,189,195,219,231,255,257,266,273,282,294,297,310,
%U A359402 313,325,334,341,350,355,365,371,381,387,397,403,413,427,443,455,471
%N A359402 Numbers whose binary expansion and reversed binary expansion have the same sum of positions of 1's, where positions in a sequence are read starting with 1 from the left.
%C A359402 Also numbers whose binary expansion and reversed binary expansion have the same sum of partial sums.
%C A359402 Also numbers whose average position of a 1 in their binary expansion is (c+1)/2, where c is the number of digits.
%C A359402 Conjecture: Also numbers whose binary expansion has as least squares fit a line of zero slope, counted by A222955.
%F A359402 A230877(a(n)) = A029931(a(n)).
%e A359402 The binary expansion of 70 is (1,0,0,0,1,1,0), with positions of 1's {1,5,6}, while the reverse positions are {2,3,7}. Both sum to 12, so 70 is in the sequence.
%t A359402 Select[Range[0,100],#==0||Mean[Join@@Position[IntegerDigits[#,2],1]]==(IntegerLength[#,2]+1)/2&]
%o A359402 (Python)
%o A359402 from functools import reduce
%o A359402 from itertools import count, islice
%o A359402 def A359402_gen(startvalue=0): # generator of terms
%o A359402     return filter(lambda n:(r:=reduce(lambda c, d:(c[0]+d[0]*(e:=int(d[1])),c[1]+e),enumerate(bin(n)[2:],start=1),(0,0)))[0]<<1==(n.bit_length()+1)*r[1],count(max(startvalue,0)))
%o A359402 A359402_list = list(islice(A359402_gen(),30)) # _Chai Wah Wu_, Jan 08 2023
%Y A359402 Binary words of this type appear to be counted by A222955.
%Y A359402 For greater instead of equal sums we have A359401.
%Y A359402 These are the indices of 0's in A359495.
%Y A359402 A030190 gives binary expansion, reverse A030308.
%Y A359402 A048793 lists partial sums of reversed standard compositions, sums A029931.
%Y A359402 A070939 counts binary digits, 1's A000120.
%Y A359402 A326669 lists numbers with integer mean position of a 1 in binary expansion.
%Y A359402 Cf. A051293, A053632, A231204, A291166, A304818, A318283, A326672, A326673, A358134, A359042.
%K A359402 nonn
%O A359402 1,3
%A A359402 _Gus Wiseman_, Jan 05 2023