c-gcc, 96 bytes

Result

WA

Code [DL]

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

stdin

00000001
01001110
01101001
11001001
11110110
10111101
00010101
10000110
10010101
01111000
10011001
10001001
11010010
00110100
11001110
00101011
11100101
01110100
10101011
00001111
11111101
00011110
01000010
11110111
10101101
00101110
10111110
11010110
00001100
00110111
01011011
00011100
10001000
10000100
11100111
01001110
00000110
00010100
00000110
00000000
01000000
11011010
00100100
10111000
00000011
00111001
00101101
00100101
00001010
11001010

stdout

01100010011010000011101000100111101110100010101010

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,s,2);a=0;for(j=t;j--;)a+=j*(j+1)==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,s,2);a=0;for(j=t;j--;)a+=j*(j+1)==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,s,2);a=0;for(j=t;j--;)a+=j*(j+1)==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,s,2);a=0;for(j=t;j--;)a+=j*(j+1)==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,s,2);a=0;for(j=t;j--;)a+=j*(j+1)==t*2;}}
                                                ^
/tmp/ccuxeMOQ.o: In function `main':
code.c:(.text.startup+0xc): warning: the `gets' function is dangerous and should not be used.

strace