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Results tagged with integer-sequences
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user 38624
For questions about sequences of integers. References are often made to the online resource oeis.org.
16
votes
Accepted
Sequences with integral means
Will Sawin's guess that the mean stabilizes around $2\sqrt{n/3}$ is very close to the correct answer, but not quite! Asymptotically the right answer is $\sim 2\sqrt{n/\pi}$. Note that $2/\sqrt{\pi}$ …
- 43.7k
14
votes
Accepted
Asymptotic behavior of the sequence $u_n = u_{n-1}^2-n$
Let us just consider the case $\alpha=2$ where there is an elegant answer: There exists a
constant $\lambda$ such that for all $n$ we have $u_n = \lceil \lambda^{2^n}\rceil$. First note that by ind …
- 43.7k
22
votes
Accepted
Are the asymptotics of A003238 known?
I'll show that
$$
\log a(n) \sim \frac{(\log n)^2}{\log 4} \approx 0.7213\ldots (\log n)^2.
$$
So the range for the constant given in the conjecture is false, but an asymptotic of that general sha …
- 43.7k
13
votes
Positive integers written as $\binom{w}2+\binom{x}4+\binom{y}6+\binom{z}8$ with $w,x,y,z\in\...
2-4-6-8, this we don't appreciate!
There is a long tradition of exploring additive representations of integers by polynomial sequences. The gold standard for measuring such progress is Waring's pro …
- 43.7k
55
votes
Accepted
A remarkable almost-identity
Consider $F(z) = 4/(3+\exp(4z))$ as a function of the complex variable $z$. It is meromorphic and has simple poles where the denominator vanishes. Namely when $4z = \log 3 + (2k +1)\pi i$ for intege …
- 43.7k