I'm looking for attributes/characteristics/properties to prove about Pascal's Triangle or the Fibonacci numbers. Preferably something that requires a strong induction proof that is on the same level as proving things such as the sum of the elements in the nth row of Pascal's triangle is $2^n$. Simple induction proof ideas are also fine.

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    $\begingroup$ How about Pascal's triangle and the Fibonacci numbers? Try proving that $F_{n+1} = \sum_{k \ge 0} {n-k \choose k}$. $\endgroup$ – Qiaochu Yuan Dec 5 '18 at 23:45
  • $\begingroup$ The known properties of Pascal's triangle, and of the Fibonacci sequence, that can be proved by induction, can each fill a library. $\endgroup$ – DanielWainfleet Dec 6 '18 at 9:11

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