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A repfigit, or Keith number, is an integer such that, when its digits start a Fibonacci sequence with that number of digits, the original number is eventually reached. An example is 47, because the Fibonacci sequence starting with 4 and 7 (4, 7, 11, 18, 29, 47) reaches 47.
They can be defined for negative indices by [3] ... The sums of the coefficients are the Fibonacci numbers. If F(n,k) is the coefficient of x k in F n (x), namely
Yuri Matiyasevich was able to show that the Fibonacci numbers can be defined by a Diophantine equation, which led to his solving Hilbert's tenth problem. [69] The Fibonacci numbers are also an example of a complete sequence. This means that every positive integer can be written as a sum of Fibonacci numbers, where any one number is used once at ...
Each rectangle has a Fibonacci number F j as width (blue number in the center) and F j−1 as height. The vertical bands have width 10. In mathematics, Zeckendorf's theorem, named after Belgian amateur mathematician Edouard Zeckendorf, is a theorem about the representation of integers as sums of Fibonacci numbers.
The negafibonacci code for a particular nonzero integer is exactly that of the integer's negafibonacci representation, except with the order of its digits reversed and an additional "1" appended to the end. The negafibonacci code for all negative numbers has an odd number of digits, while those of all positive numbers have an even number of digits.
300 to 100 BCE [10] Negative one: −1 −1 300 to 200 BCE Cube root of 2: 1.25992 10498 ... where F n is the n th Fibonacci number.
F, also called the Fibonacci factorial, where n is a nonnegative integer, is defined as the product of the first n positive Fibonacci numbers, i.e. !:= =,, where F i is the i th Fibonacci number, and 0! F gives the empty product (defined as the multiplicative identity, i.e. 1).
To encode an integer N: . Find the largest Fibonacci number equal to or less than N; subtract this number from N, keeping track of the remainder.; If the number subtracted was the i th Fibonacci number F(i), put a 1 in place i − 2 in the code word (counting the left most digit as place 0).