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Lucas numbers L(n) 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, ... L(n) = L(n − 1) + L(n − 2) for n ≥ 2, with L(0) = 2 and L(1) = 1. A000032: Prime numbers p n: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... The prime numbers p n, with n ≥ 1. A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. A000040 ...
Second edition of the book. Neil Sloane started collecting integer sequences as a graduate student in 1964 to support his work in combinatorics. [8] [9] The database was at first stored on punched cards.
Gilbreath observed a pattern while playing with the ordered sequence of prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ... Computing the absolute value of the difference between term n + 1 and term n in this sequence yields the sequence
142857 is the best-known cyclic number in base 10, being the six repeating digits of 1 / 7 (0. 142857). [2] [3] [4] [5]If 142857 is multiplied by 2, 3, 4, 5 ...
(Because 3(4k + 1) + 1 = 12k + 4 = 4(3k + 1).) In more generality: For all p ≥ 1 and odd h, f p − 1 (2 p h − 1) = 2 × 3 p − 1 h − 1. (Here f p − 1 is function iteration notation.) For all odd h, f(2h − 1) ≤ 3h − 1 / 2 The Collatz conjecture is equivalent to the statement that, for all k in I, there exists an integer n ...
An integer sequence is computable if there exists an algorithm that, given n, calculates a n, for all n > 0. The set of computable integer sequences is countable.The set of all integer sequences is uncountable (with cardinality equal to that of the continuum), and so not all integer sequences are computable.
Lucas numbers and Fibonacci numbers form complementary instances of Lucas sequences. The Lucas sequence has the same recursive relationship as the Fibonacci sequence, where each term is the sum of the two previous terms, but with different starting values. [1] This produces a sequence where the ratios of successive terms approach the golden ...
In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series.Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations on the formal series.