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All these sequences may be viewed as generalizations of the Fibonacci sequence. In particular, Binet's formula may be generalized to any sequence that is a solution of a homogeneous linear difference equation with constant coefficients. Some specific examples that are close, in some sense, to the Fibonacci sequence include:
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. A repfigit can be a tribonacci sequence if there are 3 digits in the ...
The Fibonacci sequence is frequently referenced in the 2001 book The Perfect Spiral by Jason S. Hornsby. A youthful Fibonacci is one of the main characters in the novel Crusade in Jeans (1973). He was left out of the 2006 movie version, however. The Fibonacci sequence and golden ratio are briefly described in John Fowles's 1985 novel A Maggot.
The k-Wall–Sun–Sun primes can be explicitly defined as primes p such that p 2 divides the k-Fibonacci number (()), where F k (n) = U n (k, −1) is a Lucas sequence of the first kind with discriminant D = k 2 + 4 and () is the Pisano period of k-Fibonacci numbers modulo p. [15]
That is to say, the Fibonacci sequence is a divisibility sequence. F p is prime for 8 of the first 10 primes p; the exceptions are F 2 = 1 and F 19 = 4181 = 37 × 113. However, Fibonacci primes appear to become rarer as the index increases. F p is prime for only 26 of the 1229 primes p smaller than 10,000. [3]
In the Fibonacci sequence, each number is the sum of the previous two numbers. Fibonacci omitted the "0" and first "1" included today and began the sequence with 1, 2, 3, ... . He carried the calculation up to the thirteenth place, the value 233, though another manuscript carries it to the next place, the value 377.
Another example in this chapter involves the growth of a population of rabbits, where the solution requires generating a numerical sequence. [8] Although the problem dates back long before Leonardo, its inclusion in his book is why the Fibonacci sequence is named after him today.
In 1202, Leonardo Fibonacci introduced the Fibonacci sequence to the western world with his book Liber Abaci. [5] Fibonacci presented a thought experiment on the growth of an idealized rabbit population. [6] Johannes Kepler (1571–1630) pointed out the presence of the Fibonacci sequence in nature, using it to explain the pentagonal form of ...