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Results tagged with sequences-and-series
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user 5712
for questions about sequences and series, e.g. convergence, closed form expressions, etc. Note that there is a different tag for spectral sequences, and also note that MathOverflow is not for homework. Please consider consulting the online encyclopedia for integer sequences, if you are trying to identify a given sequence that you have found in your research.
14
votes
Accepted
Generating function of the Thue-Morse sequence
Let
$$F(x)=1-x-x^2+x^3-x^4+x^5+\ldots=(1-x)(1-x^2)(1-x^4)\ldots.$$
Then $F(x)=(1-x)F(x^2)$ and
$$F(x)=1+x+x^2+x^3+\ldots-2T(x)=\frac{1}{1-x}-2T(x).$$
So
$$T(x)-(1-x)T(x^2)=\frac{x}{1-x^2}.$$
- 12.3k
11
votes
1
answer
413
views
Is Somos-8 $\mod 2$ periodic?
It is known that the Somos-$k$ sequences for $k\ge 8$ do not give integers. But the first terms of Somos-8 sequence $s_n=a_n/b_n$
$$1, 1, 1, 1, 1, 1, 1, 1, 4, 7, 13, 25, 61, 187, 775, 5827, 14815,\fra …
- 12.3k
8
votes
Does this sequence always give an integer?
Andrew Hone in the articles Analytic solutions and integrability for bilinear recurrences of order six and Sigma-function solution to the general Somos-6 recurrence via hyperelliptic Prym varieties (w …
- 12.3k
7
votes
Closed form for $\sum_{n=1}^{\infty} \frac{1}{1+n+n^2+\cdots+n^a}$
This is not an answer but a conjecture. It based on a first examples examples calculated by Mathematica.
$$S(a)=\frac1{a+1}-\sum_{i=1}^a\frac{\psi(-\alpha_i)}{f_a'(\alpha_i)},$$
where $f_0(x)=1$, $f_1 …
- 12.3k
7
votes
1
answer
278
views
On one class of Somos-like sequences
This question is motivated by integrability of the sequence mistakenly arisen in the question Does this sequence always give an integer?
Let $m_1,\ldots, m_{k-1}$ be positive integers and sequence $\ …
- 12.3k
6
votes
Accepted
Asymptotic rate for $\sum\binom{n}k^{-1}$
This sum can be written in the form, see 2-adic Logarithm and Resistance of n-dimensional Cube
$$S_{n+1}=\frac{n+1}{2^{n+1}}\sum_{k=1}^{n+1}\frac{2^k}{k}.$$
The last term in the sum gives the estimate …
- 12.3k
6
votes
The sequence $G(n,k)=G(n-2,k)+G(n,k-2)$
The sequence $G(2n,2k)$ is $T(n,k)/2$, where $T(n,k)$ is A051601. For $G(2n,2k+1)$ boundary conditions $1,3,5,7,\ldots$ can be replaced by $0,2,4,6$ (minus one Pascal triangle). It gives $G(2n,2k)$ ag …
- 12.3k
5
votes
A curious series related to the asymptotic behavior of the tetration
Let $$c_n(\lambda)=\frac{e^\lambda-{^n a}}{ \lambda^n},\quad d_n(\lambda)=e^{-\lambda}c_n(\lambda)$$ ($d_n$ seems to be simpler). In particular initial functions $c_0(\lambda)=e^\lambda-1$ and $d_0(\l …
- 12.3k
4
votes
0
answers
168
views
Positivity conjecture for Somos sequences
Let $\{s_n\}$ be the Somos-$4$ sequence, which is defined by $$s_{n+4}s_n=\alpha s_{n+3}s_{n+1}+\beta s_n^2.$$ It is known that $s_n$ is a Laurent polynomial: $s_n\in\mathbb{Z}[s_1^{\pm1}, \ldots, s_4 …
- 12.3k
4
votes
Does this sequence always give an integer?
It is not an answer just a piece of fun.
I've made a mistake in my program and calculated the sequence with wrong recurrence $$a_{n+6}=\frac{a_{n+5}\cdot a_{n+1}+a_{n+4}\cdot a_{n+2}\cdot {a_{n+3}}^ …
- 12.3k
3
votes
Closed form for $\sum_{i=1}^n{a^{i^2}}$
Some partial information.
If $g$ is a primitive root modulo prime $p$, then for $p\equiv 3
\pmod{4}$ $$\sum\limits_{x=1}^{p-1}g^{x^2}\equiv
0\pmod{p}.$$
If $p\equiv 1 \pmod{4}$ and
$$S_1=\sum\limits_{ …
- 12.3k
2
votes
Laplace's summation formula
One can find a good collection of summation formulae in Interpolation by J. F. Steffensen (1950).
§12. Laplace’s and Gauss’s Summation-Formulas
§14. Euler’s Summation-Formula
§15. Lubbock’s and Woolh …
- 12.3k