rdmd, 454 bytes

Result

WA

Code [DL]

import std.stdio;
import std.array;
import std.math;
import std.conv;

bool issq(int a) {
    return cast(int)sqrt(cast(real)a)*cast(int)sqrt(cast(real)a) == a;
}

bool tri(int a) {
    return issq(8*a+1) && (cast(int)sqrt(8.0*a+1) % 2);
}
void main() {
    string s;
    while ((s = readln()).length > 3) {
        int i = to!int(split(s)[0],2);
        if(tri(i)) {
            writeln("1");
        } else {
            writeln("0");
        }
    }
}

stdin

01010001
11100111
00101111
11010010
10101011
00100100
00000110
00001111
00100001
11111101
11011000
10011000
00110111
11000000
10010010
01001110
00000000
00011111
11101000
01001101
10111110
01011011
00110100
00010111
10001000
01000100
00011100
10000011
11110010
00101101
10001001
00000001
10111011
00010101
10011011
11100111
01111010
11111000
01000010
00001010
10100110
01000000
00001111
01101001
10011001
01111000
10110110
00110110
01100100
00000011

stdout

stderr

Error: module CODE is in file '/volume/CODE.d' which cannot be read
import path[0] = /volume
import path[1] = /usr/include/dmd/phobos
import path[2] = /usr/include/dmd/druntime/import
Failed: ["dmd", "-v", "-o-", "/volume/CODE.d", "-I/volume"]

strace