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In a similar manner it may be shown that the sum of the first Fibonacci numbers up to the n-th is equal to the (n + 2)-th Fibonacci number minus 1. [33] In symbols: = = + This may be seen by dividing all sequences summing to + based on the location of the first 2.
That is, after two starting values, each number is the sum of the two preceding numbers. The Fibonacci sequence has been studied extensively and generalized in many ways, for example, by starting with other numbers than 0 and 1, by adding more than two numbers to generate the next number, or by adding objects other than numbers.
where F n is the n th Fibonacci number. Such a sum is called the Zeckendorf representation of N. The Fibonacci coding of N can be derived from its Zeckendorf representation. For example, the Zeckendorf representation of 64 is 64 = 55 + 8 + 1. There are other ways of representing 64 as the sum of Fibonacci numbers 64 = 55 + 5 + 3 + 1 64 = 34 ...
The reciprocal Fibonacci constant ψ is the sum of the reciprocals of the Fibonacci numbers: = = = + + + + + + + +. Because the ratio of successive terms tends to the reciprocal of the golden ratio, which is less than 1, the ratio test shows that the sum converges.
As with the Fibonacci numbers, each Lucas number is defined to be the sum of its two immediately previous terms, thereby forming a Fibonacci integer sequence. The first two Lucas numbers are L 0 = 2 {\displaystyle L_{0}=2} and L 1 = 1 {\displaystyle L_{1}=1} , which differs from the first two Fibonacci numbers F 0 = 0 {\displaystyle F_{0}=0 ...
Hosoya's triangle or the Hosoya triangle (originally Fibonacci triangle; OEIS: A058071) is a triangular arrangement of numbers (like Pascal's triangle) based on the Fibonacci numbers. Each number is the sum of the two numbers above in either the left diagonal or the right diagonal.
In mathematics, the greedy algorithm for Egyptian fractions is a greedy algorithm, first described by Fibonacci, for transforming rational numbers into Egyptian fractions. An Egyptian fraction is a representation of an irreducible fraction as a sum of distinct unit fractions, such as 5 / 6 = 1 / 2 + 1 / 3 .
Fibonacci search has an average- and worst-case complexity of O(log n) (see Big O notation). The Fibonacci sequence has the property that a number is the sum of its two predecessors. Therefore the sequence can be computed by repeated addition. The ratio of two consecutive numbers approaches the Golden ratio, 1.618... Binary search works by ...