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# User:Mikhail Gaichenkov

I am from Moscow, Russia, was born in 1972. The most valuable achievement for me is the proof of the property of Kendall-Mann sequence (Reference: A00140, M(n+1)/M(n)=n-1/2+O(1/n), n--> infinity). http://mathoverflow.net/questions/46368/the-property-of-kendall-mann-numbers (Broken link: http://mathoverflow.net/questions/51324/applications-of-the-property-of-kendall-mann-numbers))

In addition to that the Products of Necklaces formula (The limit of products of the numbers of fixed necklaces of length n composed of beads of types N(n,a), n--> infinity) is a research subject for me (Reference: A008302).

*My favorite sequences.*

A000217 Triangular numbers: a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n.
a(n+1), n>=0, is the number of levels with energy **n+3/2** (in units of h*f0, with Planck's constant h and the oscillator frequency f0) of the three dimensional isotropic harmonic quantum oscillator. See the comment by A. Murthy above: n=n1+n2+n3 with positive integers and ordered. Proof from the o.g.f. See the A. Messiah reference. Wolfdieter Lang, Jun 29 2007

A186491 Counts a family of permutations occurring in the study of squeezed states of the simple harmonic oscillator.
This family of permutations have arisen in the study of squeezed states
of the simple harmonic oscillator [Sukumar and Hodges].
The sequence a(n), with the convention a(0) = 1, enumerates permutations p(1)p(2)....p(4*n) in the symmetric group on 4*n letters having the following properties:
1) The permutation can be written as a product of disjoint two cycles.
2) For i = 1,...,2*n, positions 2*i-1 and 2*i are either both ascents(labelled A) or both descents (labelled D).
The set of permutations satisfying condition (1) forms **a subgroup of Symm(4*n)** of order A001147(4*n).

A079883 a(1) = 1; a(n) = prime(n) - prime(n-1)* a(n-1) if n > 1.
Let s(n) be a sequence such that lim s(n)/s(n+1) = K different from -1. The "oscillator sequence" (or simply "oscillator") of s(n) is the sequence s'(n) defined by the rules: s'(1) = 1; s'(n) = 1 - (s(n-1)/s(n)) s'(n-1). 2. It is an open problem whether the oscillator (prime)' converges to **1/2** or diverges.

A079895 a(1) = 1; a(n) = phi(n) - phi(n-1)* a(n-1) if n > 1.
Let s(n) be a sequence such that lim s(n)/s(n+1) = K different from -1. The "oscillator sequence" (or simply "oscillator") of s(n) is the sequence s'(n) defined by the rules: s'(1) = 1; s'(n) = 1 - (s(n-1)/s(n)) s'(n-1). 2. It is an open problem whether the oscillator (prime)' converges to **1/2** or diverges.

A079897 a(1) = 1; a(n) = sigma(n) - sigma(n-1)* a(n-1) if n > 1.
Let s(n) be a sequence such that lim s(n)/s(n+1) = K different from -1. The "oscillator sequence" (or simply "oscillator") of s(n) is the sequence s'(n) defined by the rules: s'(1) = 1; s'(n) = 1 - (s(n-1)/s(n)) s'(n-1). 2. It is an open problem whether the oscillator (prime)' converges to **1/2** or diverges.

A079898 a(1) = 1; a(n) = tau(n) - tau(n-1)* a(n-1) if n > 1.
Let s(n) be a sequence such that lim s(n)/s(n+1) = K different from -1. The "oscillator sequence" (or simply "oscillator") of s(n) is the sequence s'(n) defined by the rules: s'(1) = 1; s'(n) = 1 - (s(n-1)/s(n)) s'(n-1). 2. It is an open problem whether the oscillator (prime)' converges to **1/2** or diverges.

A079899 a(1) = 1; a(n) = Fibonacci(n) - Fibonacci(n-1)* a(n-1) if n > 1.
Let s(n) be a sequence such that lim s(n)/s(n+1) = K different from -1. The "oscillator sequence" (or simply "oscillator") of s(n) is the sequence s'(n) defined by the rules: s'(1) = 1; s'(n) = 1 - (s(n-1)/s(n)) s'(n-1). 2. It is an open problem whether the oscillator (prime)' converges to **1/2** or diverges.

A213343 1-quantum transitions in systems of N spin **1/2** particles, in columns by combination indices.

A000041 a(n) = number of partitions of n (the partition numbers).
a(n) is also the number of conjugacy classes in the **symmetric group S_n** (and the number of irreducible representations of S_n).
M. Planat, Quantum 1/f Noise in Equilibrium: from Planck to Ramanujan