JavaScript (Node.js), 47 bytes, \$O\left(n^{\log_2\frac{1+\sqrt 5}2}\right)\$
f=n=>n+1>>1?[f(n>>2)[0]*2|n&1,f(n>>1)[0]]:[n,n]JavaScript (Node.js), 48 bytes
f=n=>n+1>>1?f(n>>2).map((c,i)=>c*2|n>>i&1):[n,n]↧
Answer by l4m2 for Break an integer into even- and odd-indexed bits
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