c-gcc, 94 bytes

Result

AC

Code [DL]

s[9];main(j,a,t){for(;gets(s);putchar(a+48)){t=strtol(s,a=0,2);for(j=t+1;j--;)a+=j*j+j==t*2;}}

stdin

10010101
00101110
00110111
11111101
01011011
00000110
00110000
11100111
01000001
10001000
10111011
10111110
01100110
01000010
00100100
01111000
10011001
01001010
11100011
00000011
00011100
00000001
11101101
10100000
10101011
01100111
10101111
11100101
11010010
01111011
00001111
01001110
00110101
11001000
11110111
00000000
10110001
10101011
10000010
11011100
00101101
01101001
00001010
00010101
01100010
01010101
00110111
10110011
00001001
11010000

stdout

00111101010101111001110010001011000101001111001000

stderr

/tmp/code.c:1:1: warning: data definition has no type or storage class
 s[9];main(j,a,t){for(;gets(s);putchar(a+48)){t=strtol(s,a=0,2);for(j=t+1;j--;)a+=j*j+j==t*2;}}
 ^
/tmp/code.c:1:1: warning: type defaults to ‘int’ in declaration of ‘s’ [-Wimplicit-int]
/tmp/code.c:1:6: warning: return type defaults to ‘int’ [-Wimplicit-int]
 s[9];main(j,a,t){for(;gets(s);putchar(a+48)){t=strtol(s,a=0,2);for(j=t+1;j--;)a+=j*j+j==t*2;}}
      ^
/tmp/code.c: In function ‘main’:
/tmp/code.c:1:6: warning: type of ‘j’ defaults to ‘int’ [-Wimplicit-int]
/tmp/code.c:1:6: warning: type of ‘a’ defaults to ‘int’ [-Wimplicit-int]
/tmp/code.c:1:6: warning: type of ‘t’ defaults to ‘int’ [-Wimplicit-int]
/tmp/code.c:1:23: warning: implicit declaration of function ‘gets’ [-Wimplicit-function-declaration]
 s[9];main(j,a,t){for(;gets(s);putchar(a+48)){t=strtol(s,a=0,2);for(j=t+1;j--;)a+=j*j+j==t*2;}}
                       ^
/tmp/code.c:1:31: warning: implicit declaration of function ‘putchar’ [-Wimplicit-function-declaration]
 s[9];main(j,a,t){for(;gets(s);putchar(a+48)){t=strtol(s,a=0,2);for(j=t+1;j--;)a+=j*j+j==t*2;}}
                               ^
/tmp/code.c:1:48: warning: implicit declaration of function ‘strtol’ [-Wimplicit-function-declaration]
 s[9];main(j,a,t){for(;gets(s);putchar(a+48)){t=strtol(s,a=0,2);for(j=t+1;j--;)a+=j*j+j==t*2;}}
                                                ^
/tmp/ccVWSMh6.o: In function `main':
code.c:(.text.startup+0xc): warning: the `gets' function is dangerous and should not be used.

strace