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[1] [2] Every term of the harmonic series after the first is the harmonic mean of the neighboring terms, so the terms form a harmonic progression; the phrases harmonic mean and harmonic progression likewise derive from music. [2] Beyond music, harmonic sequences have also had a certain popularity with architects. This was so particularly in the ...
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 Tower of Hanoi (also called The problem of Benares Temple[ 1] or Tower of Brahma or Lucas' Tower[ 2] and sometimes pluralized as Towers, or simply pyramid puzzle[ 3]) is a mathematical game or puzzle consisting of three rods and a number of disks of various diameters, which can slide onto any rod. The puzzle begins with the disks stacked on ...
The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, [ 1] and read on 5 December 1735 in The Saint Petersburg Academy of Sciences. [ 2] Since the problem had withstood the attacks ...
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The central binomial coefficients give the number of possible number of assignments of n -a-side sports teams from 2 n players, taking into account the playing area side. The central binomial coefficient is the number of arrangements where there are an equal number of two types of objects. For example, when , the binomial coefficient is equal ...
In mathematics, the n -th harmonic number is the sum of the reciprocals of the first n natural numbers : Starting from n = 1, the sequence of harmonic numbers begins: Harmonic numbers are related to the harmonic mean in that the n -th harmonic number is also n times the reciprocal of the harmonic mean of the first n positive integers.
1/2 + 1/4 + 1/8 + 1/16 + ⋯. First six summands drawn as portions of a square. The geometric series on the real line. 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. In summation notation ...
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