goruby, 54 bytes

Result

AC

Code [DL]

loop{a=((8*gets.to_i(2)+1)**0.5-1)/2;p a.to_i==a ?1:0}

stdin

01101001
01101101
10000000
10110111
11111101
00111101
00001111
00000110
11100110
10011001
00100110
01011110
10101001
10001000
00000001
01011010
00100100
00010100
00110111
00000001
11010011
00101101
01000000
00001010
01000001
00000011
01100100
11100111
01001110
01110000
01000010
01010111
11010010
11001101
00010000
00000111
00000000
01011011
00000000
11010111
10101011
01111000
00010101
01101111
00011100
10111110
11100000
00011101
11110010
11111100

stdout

1
0
0
0
1
0
1
1
0
1
0
0
0
1
1
0
1
0
1
1
0
1
0
1
0
1
0
1
1
0
1
0
1
0
0
0
1
1
1
0
1
1
1
0
1
1
0
0
0
0

stderr

/volume/CODE:1:in `to_i': wrong number of arguments (given 1, expected 0) (ArgumentError)
	from /volume/CODE:1:in `block in <main>'
	from /volume/CODE:1:in `loop'
	from /volume/CODE:1:in `<main>'

strace