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The Pell numbers have P n = 2P n−1 + P n−2. If the coefficient of the preceding value is assigned a variable value x, the result is the sequence of Fibonacci polynomials. Not adding the immediately preceding numbers. The Padovan sequence and Perrin numbers have P(n) = P(n − 2) + P(n − 3).
An example is 47, because the Fibonacci sequence starting with 4 and 7 (4, 7, 11, 18, 29, 47) reaches 47. A repfigit can be a tribonacci sequence if there are 3 digits in the number, a tetranacci number if the number has four digits, etc.
The sequence also has a variety of relationships with the Fibonacci numbers, like the fact that adding any two Fibonacci numbers two terms apart in the Fibonacci sequence results in the Lucas number in between. [3] The first few Lucas numbers are 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, ... .
A prime divides if and only if p is congruent to ±1 modulo 5, and p divides + if and only if it is congruent to ±2 modulo 5. (For p = 5, F 5 = 5 so 5 divides F 5) . Fibonacci numbers that have a prime index p do not share any common divisors greater than 1 with the preceding Fibonacci numbers, due to the identity: [6]
Plot of the first 10,000 Pisano periods. In number theory, the nth Pisano period, written as π (n), is the period with which the sequence of Fibonacci numbers taken modulo n repeats.
For n = 1, 2, 3 it is clearly true (as these are Fibonacci numbers), for n = 4 we have 4 = 3 + 1. If n is a Fibonacci number then there is nothing to prove. Otherwise there exists j such that F j < n < F j + 1 .
Lucas sequences are used in some primality proof methods, including the Lucas–Lehmer–Riesel test, and the N+1 and hybrid N−1/N+1 methods such as those in Brillhart-Lehmer-Selfridge 1975. [ 4 ] LUC is a public-key cryptosystem based on Lucas sequences [ 5 ] that implements the analogs of ElGamal (LUCELG), Diffie–Hellman (LUCDIF), and RSA ...
The nth element of a Fibonacci word, , is 1 if the Zeckendorf representation (the sum of a specific set of Fibonacci numbers) of n includes a 1, and 0 if it does not include a 1. The digits of the Fibonacci word may be obtained by taking the sequence of fibbinary numbers modulo 2. [10]