Upper or Lower Wythoff?
JavaScript (ES6), 50 35 bytes



f=(n,s="1",t=0)=&gt;s[n-1]||f(n,s+t,s)
&lt;input type=number min=1 oninput=o.textContent=this.value&amp;amp;&amp;amp;f(this.value)&gt;&lt;pre id=o&gt;




Outputs 1 for lower and 0 for upper. Explanation: Partial lists of boolean values can be constructed using a Fibonacci-like identity: given two lists, starting with 1 and 10, each subsequent list is the concatenation of the previous two, resulting in 101, 10110, 10110101 etc. In this case it's slightly golfier to have a fake 0th entry of 0 and use that to construct the second element of the list.