Find a sequence in the binary digits of π
JavaScript (Node.js),  167  165 bytes
This is long and slow ... but it should work for any input in theory.

s=&gt;{for(k=t=1n,l=q=2n,n=0n,r=6n,o='#';!~(i=o.search(s));)n=4n*q+t*~n&lt;r?(o+=n,q+(q+=q)-r)/t-n&lt;&lt;(r+=r+t*n*2n)/r:(q*7n*k+2n-r*++l)/(r=(r-q-q)*l,q*=k++,t*=l++);return i}

Try it online!
This is derived from the C# entry on this Rosetta Code page.
The C# code is derived from Java, which is derived from Icon, which is derived from PicoLisp, which is derived from Haskell, which is based on this paper. :-p