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"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.
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.
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 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 2 in the third term represents the length 2 of the block of 1s that are in the first and second term. At this point, the sequence decreases for the first time: The 1 in the fourth term represents the length 1 of the block of 2s in the 3rd term, as well as the length 1 of the block "1, 2" spanning the second and third term.
The sequence of the number 1 followed by the prime numbers (studied by S. S. Pillai [3] and others); this follows from Bertrand's postulate. [1] The sequence of practical numbers which has 1 as the first term and contains all other powers of 2 as a subset. [4] (sequence A005153 in the OEIS)