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where f (2k−1) is the (2k − 1)th derivative of f and B 2k is the (2k)th Bernoulli number: B 2 = 1 / 6 , B 4 = − + 1 / 30 , and so on. Setting f ( x ) = x , the first derivative of f is 1, and every other term vanishes, so [ 15 ]
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: Partition numbers P n: 1, 1, 2, 3, 5, 7, 11, 15, 22, 30 ...
A bijection with the sums to n is to replace 1 with 0 and 2 with 11. The number of binary strings of length n without an even number of consecutive 0 s or 1 s is 2F n. For example, out of the 16 binary strings of length 4, there are 2F 4 = 6 without an even number of consecutive 0 s or 1 s—they are 0001, 0111, 0101, 1000, 1010, 1110. There is ...
(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 nth term: an explicit definition.
3. Subfactorial: if n is a positive integer, !n is the number of derangements of a set of n elements, and is read as "the subfactorial of n". * Many different uses in mathematics; see Asterisk § Mathematics. | 1. Divisibility: if m and n are two integers, means that m divides n evenly. 2. In set-builder notation, it is used as a separator ...
Among the 22 partitions of the number 8, there are 6 that contain only odd parts: 7 + 1; 5 + 3; 5 + 1 + 1 + 1; 3 + 3 + 1 + 1; 3 + 1 + 1 + 1 + 1 + 1; 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1; Alternatively, we could count partitions in which no number occurs more than once. Such a partition is called a partition with distinct parts. If we count the ...
[1] The approximation can be proven several ways, and is closely related to the binomial theorem . By Bernoulli's inequality , the left-hand side of the approximation is greater than or equal to the right-hand side whenever x > − 1 {\displaystyle x>-1} and α ≥ 1 {\displaystyle \alpha \geq 1} .
The sequence starts with a unary operation (the successor function with n = 0), and continues with the binary operations of addition (n = 1), multiplication (n = 2), exponentiation (n = 3), tetration (n = 4), pentation (n = 5), etc. Various notations have been used to represent hyperoperations.