We need to check whether the given number is of the form:
k(k+1)/2+k+2
just consider the quadratic in k and check if
9-4*(4-2*N) is a perfect square.

#include <stdio.h>
#include <math.h>
#include <string>

using namespace std;

char solve( int n ){
if ( n==1 ) return '1';

double w;
double N=n;

w=int( sqrt( double( 9.0-4.0*(4.0-2.0*N) ) )+0.0000001 );
if ( w*w==9.0-4.0*(4.0-2.0*N) ) return '1';
return '0';
}

int main( void ){
int n,k;

scanf( "%d", &n );
while ( n-- ){
 scanf( "%d", &k );
 putc( solve(k), stdout );
 putc( ' ', stdout );
}

}

why my program dont work ?

  var
    a               : array [1..65536] of longint;
    n,i,k           : longint;
  begin
    read (n);
    for i := 1 to n do
      read (a[i]);
    for i := 1 to n do
      begin
        k := round(sqrt((a[i]-1)*2));
        if (k*(k+1)) div 2 = a[i] - 1 then write ('1 ') else write
('0 ');
      end;
  end.

> We need to check whether the given number is of the form:
> k(k+1)/2+k+2
> just consider the quadratic in k and check if
> 9-4*(4-2*N) is a perfect square.
>
> #include <stdio.h>
> #include <math.h>
> #include <string>
>
> using namespace std;
>
> char solve( int n ){
> if ( n==1 ) return '1';
>
> double w;
> double N=n;
>
> w=int( sqrt( double( 9.0-4.0*(4.0-2.0*N) ) )+0.0000001 );
> if ( w*w==9.0-4.0*(4.0-2.0*N) ) return '1';
> return '0';
> }
>
> int main( void ){
> int n,k;
>
> scanf( "%d", &n );
> while ( n-- ){
>  scanf( "%d", &k );
>  putc( solve(k), stdout );
>  putc( ' ', stdout );
> }
>
> }
>

> #include <stdio.h>
> #include <math.h>
> #include <string>
>
> using namespace std;
>
> char solve( int n ){
> if ( n==1 ) return '1';
>
> double w;
> double N=n;
>
> w=int( sqrt( double( 9.0-4.0*(4.0-2.0*N) ) )+0.0000001 );
> if ( w*w==9.0-4.0*(4.0-2.0*N) ) return '1';
> return '0';
> }
>
> int main( void ){
> int n,k;
>
> scanf( "%d", &n );
> while ( n-- ){
>  scanf( "%d", &k );
>  putc( solve(k), stdout );
>  putc( ' ', stdout );
> }
>
> }
>

/*My problem don't work too.WHY!!             */
/*SOS!!!*/
#include<iostream>
#include<vector>
using namespace std;
int main(){
    long i,k,ar[500],t;
    vector<int> num;
    vector<int>::iterator it;
    for(i=0;i<500;i++) ar[i]=i*(i+1)/2+1;
    cin>>k;
    while(k--) {cin>>t;num.insert(num.end(),t); }
    it=num.begin();
    while(it!=num.end()){
        t=*it++;i=500;
        while(t<ar[--i]);
        if(t==ar[i]) cout<<'1'<<' ';
        else cout<<'0'<<' ';
    }
    return 0;
}