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Results tagged with integer-sequences
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user 297
For questions about sequences of integers. References are often made to the online resource oeis.org.
21
votes
Is the sequence $a_n=c a_{n-1} - a_{n-2}$ always composite for $n > 5$?
Put $u = (c + \sqrt{c^2-4})/2$. We have
$$a_{2n} = \frac{u^{2n}-u^{-2n}}{u-u^{-1}} = \left( \frac{ u^n-u^{-n}}{u-u^{-1}} \right) \left( \vphantom{\frac{ u^n-u^{-n}}{u-u^{-1}}} u^n + u^{-n} \right)$$
…
- 156k
10
votes
A possible surprise involving Euler's constant $e$
Sure.
$$c_n = \frac{1}{n+1} + \frac{1}{(n+1)(n+2)}+ \frac{1}{(n+1)(n+2)(n+3)} \cdots$$
so
$$\frac{1}{n+1} < c_n < \frac{1}{n+1}+\frac{1}{(n+1)^2}+\frac{1}{(n+1)^3}+\cdots = \frac{1}{n}.$$
That prov …
- 156k
4
votes
Accepted
On one class of Somos-like sequences
I believe this is a special case of Case (9) in Theorem 3.9 of Allman, Cuenca and Huang. By the way, this paper was an REU project!
- 156k