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Is it possible to mathematically deduce the next element (to the right) in the following series? It continues in the same pattern to the left ($n-1$ copies of the positive integer $n$ on the left).
$ \ldots, 7,6,6,6,6,6,5,5,5,5,4,4,4,3,3,2,?$

Is there a most natural continuation (analytical or smooth or nice) possibly of several or infinite elements?

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    So if you're looking for a formula, try $\lceil \frac{1}{2} (\sqrt{1+8n}+1) \rceil$.2011-12-05

2 Answers 2

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The answer is "$42$".

We get a lot of these "what's the next number in the sequence" questions. And people often complain that an answer will depend on some "human emotion" about what constitutes a "natural next number." Mathematics is agnostic to such attributes, so we could put any number we like as the next one, and it would be perfectly "correct."

So I proclaim the answer is "$42$". Or, if you want something perhaps less arbitrary-feeling, then how about

$ \ldots, 5,5,5,5,4,4,4,3,3,2,0,-1,-1,-2,-2,-2,\ldots $

the rule here being to write $|n-1|$ copies of $n$ in descending order.

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    @N.S.: Congrats, you managed to break StackExchange's layout. In the future, may I suggest using `$P(1)=2$, $P(2)=3$` etc. instead of putting everything inside a single pair of dollar signs.2011-12-05
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As m. k. points out this is given in A003057 (mirror imaged) with the explicit formula $a(n)=1+\lceil\frac{\sqrt{1+8n}}{2}\rceil$

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    I understood the OP wanted a continuation where he wrote the '?' on the right.2011-12-06