Fibonacci numbers as sums of binomial coefficients

Builds on Binomial coefficient Fibonacci sequence

Statement

Let $n$ be a natural number, and let $F_m$ be the Fibonacci numbers. Then $F_{n+1} = \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n - k}{k}.$

Remarks

Example: $F_6 = \binom{5}{0} + \binom{4}{1} + \binom{3}{2} = 1 + 4 + 3 = 8$ . Combinatorial proof: $F_{n+1}$ counts the tilings of a strip of length $n$ by squares and dominoes. A tiling with $k$ dominoes has $n - k$ tiles, and $\binom{n-k}{k}$ choices fix the positions of the dominoes. In Pascal's triangle, the sum runs along a shallow diagonal.

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Binomial coefficient definition
Fibonacci sequence example
Fibonacci numbers as sums of binomial coefficients statement

Tag	`0V4VB6MQDC`
URI	`m2://0V4VB6MQDC`
Kind	statement, theorem
Writing	W0 stub
MSC	11B39
Status	This copy of the site has no build data. Run `m2 status 0V4VB6MQDC`.
Uses	Binomial coefficient, Fibonacci sequence

Fibonacci numbers as sums of binomial coefficients

Statement

Remarks

Reading path

Further reading

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