simula, 191 bytes

Result

AC

Code [DL]

begin integer a,b,r,i,c;while i<50do begin a:=inint;b:=0;r:=1;while a>0do begin b:=b+r*(a-a//10*10);r:=r*2;a:=a/10;end;a:=sqrt(b*8+1);if a*a=b*8+1then c:=1else c:=0;outint(c,0);i:=i+1;end;end

stdin

00000011
00111011
00111110
00101101
10000011
01000010
11000001
01100101
11100010
00001010
00000001
00001110
11010101
11111101
10010011
00110100
11110000
00000000
01110110
01101001
10100111
10000001
11100111
01111100
00001101
00100100
10101110
01011011
11001000
00011100
00001111
10001101
00011100
11011101
11000000
01001110
10111110
10011001
11011010
10101011
00010101
01111000
00110111
00000110
10001000
11010010
00111111
00110111
11110110
10100000

stdout

10010100011001000101001001010110100111011111110100

stderr

Compiling /tmp/code.sim:
gcc -g -O2 -c code.c
code.c: In function ‘main’:
code.c:116:136: warning: implicit declaration of function ‘exit’ [-Wimplicit-function-declaration]
 int main(int __argc, char *__argv[]){__progname= __argv[0];goto __sw;__l0:__rstart(__argc,__argv);goto __start;__l1:__slutt:__rslutt();exit(0);__start:__curent_map=__mapmain;
                                                                                                                                        ^
code.c:116:136: warning: incompatible implicit declaration of built-in function ‘exit’
code.c:116:136: note: include ‘<stdlib.h>’ or provide a declaration of ‘exit’
/tmp/code.sim:1:979: warning: implicit declaration of function ‘__rbe’ [-Wimplicit-function-declaration]
/tmp/code.sim: In function ‘__init’:
/tmp/code.sim:4:19: warning: implicit declaration of function ‘__init_FILE’ [-Wimplicit-function-declaration]
/tmp/code.sim: At top level:
/tmp/code.sim:5:1: warning: return type defaults to ‘int’ [-Wimplicit-int]
/tmp/code.sim:13:1: warning: return type defaults to ‘int’ [-Wimplicit-int]
gcc -g -O2 -o /tmp/code code.o -L/usr/local/lib -lcim

strace