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Let $(a_n)$ be the sequence of minimum values of the expression (depending on $\ell$):

\begin{equation*} a_n=\arg\min \bigg\lbrace ((\ell+1)j-n)!\,(n-\ell j)!\quad \text{for}\; \,\bigg\lceil\frac{n}{\ell+1}\bigg\rceil\le j\le \bigg\lfloor\frac{n}{\ell}\bigg\rfloor \bigg\rbrace % \min \big\lbrace (2j-n)!\,(n- j)!,\; \mathrm{ceil}\,\frac{n}{2}\le j\le n \big\rbrace, .\end{equation*}

My questions is two fold:

Is there a closed-form expression for $(a_n)_{n\ge \ell}$ (for a fixed $\ell$), or a way to characterize its entries? Why?

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    So, $l$ is fixed in the definition of $a_n$, and only $j$ varies?2012-08-08
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    Yes, that is correct.2012-08-08
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    For instance, for $\ell=1$, the sequence $(a_n)$ is: \begin{align*} 1,& 1, 2, 2, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 9, 9, 10, 10, 11, 11,\\ 12, &12, 13, 13, 14, 14, 15, 16, 16, 17, 17, 18, 18,\\ 19, &19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 25, 25, 26, 26,\\ 27, &27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32,... \end{align*} The patterns appears to be: Two instances of every integer, except integers of the form $(n+1)^2-1$ which occurs only once in the sequence.2012-08-08
  • 0
    You've checked the OEIS already?2012-08-09
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    Yes, with no luck.2012-08-09
  • 1
    You mean *closeD formula*.2012-08-09
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    @dbir The values you give for $a_n$ above look more like values of $j$. Did you mean "arg min" rather than "min"? Are there possibly ties? If so, how are you defining $j$ to be chosen?2012-08-09
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    Yes, I meant closed formula. I apologize.2012-08-09
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    It is "[closed-form expression](http://en.wikipedia.org/wiki/Closed-form_expression)", not _closed formula_".2012-08-09
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    Also posted, without acknowledgement here or there, to MO: http://mathoverflow.net/questions/104367/a-closed-formula-for-a-sequence-of-integers2012-08-09
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    Erick is correct, I meant the argument of the minimum. Respect to the question about possible ties and what $j$ to choose, I do not know the answer, but any sequence that does the job is more than welcome.2012-08-10
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    I also posted (a similar) question in MathOverflow: http://mathoverflow.net/questions/104367/a-closed-formula-for-a-sequence-of-integers2012-08-10
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    There's an echo in here.2012-08-10

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