Recurrence Relation for Sequence of mth Powers of Fibonacci Numbers/Examples/m = 3
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Example of Recurrence Relation for Sequence of mth Powers of Fibonacci Numbers
- ${F_n}^n - 2 {F_{n + 1} }^2 - 2 {F_{n + 2} }^2 + {F_{n + 3} }^2 = 0$
Proof
Set $m = 3$ into Recurrence Relation for Sequence of mth Powers of Fibonacci Numbers:
- $\ds \sum_{k \mathop \in \Z} \dbinom m k_\FF \paren {-1}^{\ceiling {\paren {m - k} / 2} } {F_{n + k} }^{m - 1} = 0$
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.8$: Fibonacci Numbers: Exercise $30$