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If one traces the pedigree of any male bee (1 bee), he has 1 parent (1 bee), 2 grandparents, 3 great-grandparents, 5 great-great-grandparents, and so on. This sequence of numbers of parents is the Fibonacci sequence.
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.
"subtract if possible, otherwise add": a(0) = 0; for n > 0, a(n) = a(n − 1) − n if that number is positive and not already in the sequence, otherwise a(n) = a(n − 1) + n, whether or not that number is already in the sequence.
(the Fibonacci sequence) is formed by starting with 0 and 1 and then adding any two consecutive terms to obtain the next one: an implicit description (sequence A000045 in the OEIS). The sequence 0, 3, 8, 15, ... is formed according to the formula n 2 − 1 for the n th term: an explicit definition.
In mathematics, the Fibonacci numbers form a sequence defined recursively by: = {= = + > That is, after two starting values, each number is the sum of the two preceding numbers.
The look-and-say sequence is also popularly known as the Morris Number Sequence, after cryptographer Robert Morris, and the puzzle "What is the next number in the sequence 1, 11, 21, 1211, 111221?" is sometimes referred to as the Cuckoo's Egg , from a description of Morris in Clifford Stoll 's book The Cuckoo's Egg .
As another example, π is the limit of the sequence (3, 3.1, 3.14, 3.141, 3.1415, ...), which is increasing. A related sequence is the sequence of decimal digits of π, that is, (3, 1, 4, 1, 5, 9, ...). Unlike the preceding sequence, this sequence does not have any pattern that is easily discernible by inspection.
In the third movement of Bartók's Music for Strings, Percussion and Celesta, the opening xylophone passage uses Fibonacci rhythm as such: 1:1:2:3:5:8:5:3:2:1:1. [ 10 ] The Fibonacci numbers are also apparent in the organisation of the sections in the music of Claude Debussy 's Image, Reflections in Water , in which the sequence of keys is ...