Convergence of sequence

Compute values of the sequence [math]\displaystyle{ (a_n)_{n\in{\mathbb N}} }[/math].

nth-element of the sequence:
start at

The underlying JavaScript code

 var board = JXG.JSXGraph.initBoard('box', {axis:true, boundingbox: [-3, 8, 50, -8]});
 var seq = board.create('curve', [[0], [1]], {strokeColor: 'blue'});
 
 var seq_add = function() {
    var n = seq.dataX.length,
        val = a_n(n);
    seq.dataX.push(n);
    seq.dataY.push(val);
 };

var txt1 = board.create('text', [15, 1.6, function() { return 'n=' + (seq.dataX.length-1) + ': value = ' + seq.dataY[seq.dataY.length - 1]; }], {strokeColor: 'blue'});

var TO;

var approx = function() {
     seq_add();
     board.update();
     if (seq.dataX.length <= 50) {
         TO = setTimeout(approx, 500);
     }
};

var a_n;
var start_approx = function() {
  var txtraw = document.getElementById('input').value;
  a_n = board.jc.snippet(txtraw, true, 'n', true);
  approx();
}

var clear_all = function() {
  clearTimeout(TO);
  seq.dataX = [0];
  seq.dataY = [1];
};