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nxt[{a_, b_}]:={b, 1/a+1/b}; NestList[nxt, {1, 2}, 20][[;; , 1]]//Denominator (* Harvey P. Dale, Apr 02 2024 *)
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a(n) is the numerator of b(n) where b(n)=1/(b(n-1)+ b(n-2)) with b(1)=1 and b(2)=2 . [Mark Dols, Jul 17 2009]
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