Descrete Fourier coefficient determination

I want to develop a Fourier coefficients determination algorithm. I have a challenge of creating one because whenever I increase the Fourier Harmonics the first Fourier coefficients remain the same and I only see the addition of the new one.

The simulated values do not make sense all after a certain harmonics. How can I fix this. I attached my code.

for example, for the below set of points I determined the Fourier approximation:

 xx=[-3.14;-2.44;-1.75;-1.05;-0.35;0.35;1.05;1.75;2.44;3.14];
 yy=[-8.87;-4.97;-2.05;-0.10;0.88;0.88;-0.10;-2.05;-4.97;-8.87];

For N=4

 an=  n=0,1,2..
    -2.3722
    4.1685
   -1.1708
    0.6241
   -0.4856

For N=13

 an=
   -2.3722
    4.1685
   -1.1708
    0.6241
   -0.4856
    0.5250
   -0.7055
    1.1980
   -4.1518
    4.7429
   -4.1841
    1.1457
   -0.5443
    0.4454

*

[xnt,ynt]=FR_approximation(xx,yy,N);

% xx=[-3.14;-2.44;-1.75;-1.05;-0.35;0.35;1.05;1.75;2.44;3.14];
% 
% yy=[-8.87;-4.97;-2.05;-0.10;0.88;0.88;-0.10;-2.05;-4.97;-8.87];
% 
% N=13;

% The fourier descriptors calculation

    an=[ ];  bn=[ ];     % initialization of Fourier coefficients

% determining the first term of the series a0

n=length(xx);    

deltax=2*pi/(n-1);   beta= (2*pi)/(xx(end)-xx(1));

% Calcuklating a0

      suma0=[ ]; 

      for i=1:n-1

          suma_0=yy(i)*deltax;                                % part of the calculation of an
          suma0=[suma0;suma_0];

       end

      a0=(1/(2*pi))*sum(suma0)

     % the number of intervals

for k=1:N    % N is the number of harmonics

      suma=[ ]; sumb=[ ];     % initialization

      for i=1:n-1

          suma1=yy(i)*cos(k*beta*xx(i))*deltax;                                % part of the calculation of an
          suma=[suma;suma1];

          sumb1=yy(i)*sin(k*beta*xx(i))*deltax;
          sumb=[sumb;sumb1];                                          % part of the calculation of bn
      end

      ani=(1/(pi))*sum(suma);
      bni=(1/(pi))*sum(sumb);

      %
      an=[an;ani];                                                  % Fourier coefficients 
      bn=[bn;bni];

end