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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.
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
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
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
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