Digital Sum Fibonacci
JavaScript (ES6), 45 bytes



f=(n,x=0,y=1)=&gt;n?f(n-1,y,(x%9||x)+(y%9||y)):x
&lt;input type=number min=0 oninput=o.textContent=f(this.value)&gt;&lt;pre id=o&gt;




x and y can't both be 9, since that would require the previous number to be 0, so their digital sum can't exceed 17. This means that the digital root for numbers greater than 9 is the same as the remainder modulo 9.