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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 ]
Lucas numbers have L 1 = 1, L 2 = 3, and L n = L n−1 + L n−2. Primefree sequences use the Fibonacci recursion with other starting points to generate sequences in which all numbers are composite. Letting a number be a linear function (other than the sum) of the 2 preceding numbers. The Pell numbers have P n = 2P n−1 + P n−2.
The unary numeral system is the simplest numeral system to represent natural numbers: [1] to represent a number N, a symbol representing 1 is repeated N times. [2]In the unary system, the number 0 (zero) is represented by the empty string, that is, the absence of a symbol.
21 is read off as "one 2, one 1" or 1211. 1211 is read off as "one 1, one 2, two 1s" or 111221. 111221 is read off as "three 1s, two 2s, one 1" or 312211. The look-and-say sequence was analyzed by John Conway [1] after he was introduced to it by one of his students at a party. [2] [3]
123 is a Lucas number. [1] It is the eleventh member of the Mian–Chowla sequence. [2] Along with 6, 123 is one of only two positive integers that is simultaneously two more than a perfect square and two less than a perfect cube (123 = 11 2 + 2 = 5 3 - 2). [3] 123 is the first whole number containing numbers from 1 to 3.
In mathematics, the infinite series 1 / 2 + 1 / 4 + 1 / 8 + 1 / 16 + ··· is an elementary example of a geometric series that converges absolutely. The sum of the series is 1.
For some other divergent geometric series, including Grandi's series with ratio −1, and the series 1 + 2 + 4 + 8 + ⋯ with ratio 2, one can use the general solution for the sum of a geometric series with base 1 and ratio , obtaining , but this summation method fails for 1 + 1 + 1 + 1 + ⋯, producing a division by zero.
CD-ROM encyclopedias were usually a macOS or Microsoft Windows (3.0, 3.1 or 95/98) application on a CD-ROM disc. The user would execute the encyclopedia's software program to see a menu that allowed them to start browsing the encyclopedia's articles, and most encyclopedias also supported a way to search the contents of the encyclopedia.