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

1,3

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.

Links

Table of n, a(n) for n=1..58.

Formula

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

Example

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.

Mathematica

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

Prog

(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

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.

Sequence in context: A376857 A342572 A374199 * A329358 A180204 A006995

Adjacent sequences: A359399 A359400 A359401 * A359403 A359404 A359405

Keyword

nonn

Author

Gus Wiseman, Jan 05 2023

Status

approved