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Here is my favorite: the Zeckendorf family identities (by Philip Matchett Wood and Doron Zeilberger).

Every sufficiently high (= high enough for the right hand sides to make sense) integer $n$ satisfies

$1f_n=f_n$;

$2f_n=f_{n-2}+f_{n+1}$;

$3f_n=f_{n-2}+f_{n+2}$;

$4f_n=f_{n-2}+f_n+f_{n+2}$;

$5f_n=f_{n-4}+f_{n-1}+f_{n+3}$;

etc.

The pattern behind these identities is: $kf_n$ on the left, a sum of $f_{n+\alpha}$ on the right, where no $\alpha$ occurs twice, and no two consecutive integers both occur as $\alpha$'s.

It turns out that such an identity is unique for all $k$.

(Shameless plug:) It generalizes.

1

Here is my favorite: the Zeckendorf family identities (by Philip Matchett Wood and Doron Zeilberger).

Every sufficiently high (= high enough for the right hand sides to make sense) integer $n$ satisfies

$1f_n=f_n$;

$2f_n=f_{n-2}+f_{n+1}$;

$3f_n=f_{n-2}+f_{n+2}$;

$4f_n=f_{n-2}+f_n+f_{n+2}$;

$5f_n=f_{n-4}+f_{n-1}+f_{n+3}$;

etc.

The pattern behind these identities is: $kf_n$ on the left, a sum of $f_{n+\alpha}$ on the right, where no $\alpha$ occurs twice, and no two consecutive integers both occur as $\alpha$'s.

It turns out that such an identity is unique for all $k$.

(Shameless plug:) It generalizes.