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0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 62, ... "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.
For example, P(6) = 4, and there are 4 ways to write 11 as an ordered sum in which each term is odd and greater than 1: 11 ; 5 + 3 + 3 ; 3 + 5 + 3 ; 3 + 3 + 5. The number of ways of writing n as an ordered sum in which each term is congruent to 2 mod 3 is equal to P(n − 4). For example, P(6) = 4, and there are 4 ways to write 10 as an ordered ...
In words: the first two numbers in the sequence are both 2, and each successive number is formed by adding twice the previous Pell–Lucas number to the Pell–Lucas number before that, or equivalently, by adding the next Pell number to the previous Pell number: thus, 82 is the companion to 29, and 82 = 2 × 34 + 14 = 70 + 12.
Illustration of the filling of the unit interval (horizontal axis) using the first n terms of the decimal Van der Corput sequence, for n from 0 to 999 (vertical axis). A van der Corput sequence is an example of the simplest one-dimensional low-discrepancy sequence over the unit interval; it was first described in 1935 by the Dutch mathematician J. G. van der Corput.
One can then prove that this smoothed sum is asymptotic to − + 1 / 12 + CN 2, where C is a constant that depends on f. The constant term of the asymptotic expansion does not depend on f: it is necessarily the same value given by analytic continuation, − + 1 / 12 . [1]
Mathematics – Poker: The odds of being dealt a royal flush in poker are 649,739 to 1 against, for a probability of 1.5 × 10 −6 (0.000 15%). [9] Mathematics – Poker: The odds of being dealt a straight flush (other than a royal flush) in poker are 72,192 to 1 against, for a probability of 1.4 × 10 −5 (0.0014%).
A more complicated formula, giving the same value, but generalizing more readily to higher dimensions, is: [1] (,) = (). In this formula, squaring and then taking the square root leaves any positive number unchanged, but replaces any negative number by its absolute value. [1]
A list of articles about numbers (not about numerals). Topics include powers of ten, notable integers, prime and cardinal numbers, and the myriad system.