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In mathematics, the Fibonacci sequence is a sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers , commonly denoted F n .
"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.
A Fibonacci sequence of order n is an integer sequence in which each sequence element is the sum of the previous elements (with the exception of the first elements in the sequence). The usual Fibonacci numbers are a Fibonacci sequence of order 2.
Prime number – Number divisible by only 1 or itself Prime number theorem; Distribution of primes; Composite number – Number made of two smaller integers; Factor – A number that can be divided from its original number to get a whole number Greatest common factor – Greatest factor that is common between two numbers
As a second example, for sequences in the real numbers, weak positivity (is for all ?) reduces to positivity of the sequence (because the answer must be negated, this is a Turing reduction). The Skolem-Mahler-Lech theorem would provide answers to some of these questions, except that its proof is non-constructive .
For example, the Zeckendorf representation of 64 is 64 = 55 + 8 + 1. There are other ways of representing 64 as the sum of Fibonacci numbers 64 = 55 + 5 + 3 + 1 64 = 34 + 21 + 8 + 1 64 = 34 + 21 + 5 + 3 + 1 64 = 34 + 13 + 8 + 5 + 3 + 1. but these are not Zeckendorf representations because 34 and 21 are consecutive Fibonacci numbers, as are 5 and 3.
The first: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728 (sequence A000578 in the OEIS). A perfect power has a common divisor m > 1 for all multiplicities (it is of the form a m for some a > 1 and m > 1).
In 1953, Leo Moser proved that there is a single infinite Salem–Spencer sequence achieving the same asymptotic density on every prefix as Behrend's construction. [1] By considering the convex hull of points inside a sphere, rather than the set of points on a sphere, it is possible to improve the construction by a factor of log n ...