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The octagonal number for n is given by the formula 3n 2 − 2n, with n > 0. The first few octagonal numbers are 1, 8, 21, 40, 65, 96, 133, 176, 225, 280, 341, 408, 481, 560, 645, 736, 833, 936 (sequence A000567 in the OEIS) The octagonal number for n can also be calculated by adding the square of n to twice the (n − 1)th pronic number.
1, 1, 2, 2, 4, 2, 6, 4, 6, 4, ... φ(n) is the number of positive integers not greater than n that are coprime with n. A000010: 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 ...
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
The partial sums of the series 1 + 2 + 3 + 4 + 5 + 6 + ⋯ are 1, 3, 6, 10, 15, etc.The nth partial sum is given by a simple formula: = = (+). This equation was known ...
In number theory, Ramanujan's sum, usually denoted c q (n), is a function of two positive integer variables q and n defined by the formula = (,) =,where (a, q) = 1 means that a only takes on values coprime to q.
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In Jan. 2023, I wrote about my 10 top stocks to buy for the new year. I ended up pretty proud of my list because if you'd invested $1,000 in each of the 10 stocks the day the article was published ...
The regular map {6,4} 3 or {6,4} (4,0) can be seen as a 4-coloring on the {6,4} tiling. It also has a representation as a petrial octahedron , {3,4} π , an abstract polyhedron with vertices and edges of an octahedron , but instead connected by 4 Petrie polygon faces.