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.

0, 1, 3, 5, 7, 9, 15, 17, 21, 27, 31, 33, 45, 51, 63, 65, 70, 73, 78, 85, 93, 99, 107, 119, 127, 129, 150, 153, 165, 189, 195, 219, 231, 255, 257, 266, 273, 282, 294, 297, 310, 313, 325, 334, 341, 350, 355, 365, 371, 381, 387, 397, 403, 413, 427, 443, 455, 471

Offset: 1

Author

Gus Wiseman, Jan 05 2023

Keywords

• nonn

Comments

Also numbers whose binary expansion and reversed binary expansion have the same sum of partial sums.

Also numbers whose average position of a 1 in their binary expansion is (c+1)/2, where c is the number of digits.

Conjecture: Also numbers whose binary expansion has as least squares fit a line of zero slope, counted by A222955.

Examples

			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.
		

Crossrefs

Binary words of this type appear to be counted by A222955.

For greater instead of equal sums we have A359401.

These are the indices of 0's in A359495.

A030190 gives binary expansion, reverse A030308.

A048793 lists partial sums of reversed standard compositions, sums A029931.

A070939 counts binary digits, 1's A000120.

A326669 lists numbers with integer mean position of a 1 in binary expansion.

Cf. A051293, A053632, A231204, A291166, A304818, A318283, A326672, A326673, A358134, A359042.

Programs

Mathematica

Select[Range[0,100],#==0||Mean[Join@@Position[IntegerDigits[#,2],1]]==(IntegerLength[#,2]+1)/2&]

Python

from functools import reduce
from itertools import count, islice
def A359402_gen(startvalue=0): # generator of terms
    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)))
A359402_list = list(islice(A359402_gen(),30)) # Chai Wah Wu, Jan 08 2023

Formula

A230877(a(n)) = A029931(a(n)).