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I have the following sequence and I need to find a rule for thr $n^{th}$ term: $$\lbrace a_n\rbrace=\lbrace 0,1,1,2,2,3,3,4,4...\rbrace$$ The rule that I thought about is: $$a_n=\left\lfloor\frac{n}2\right\rfloor, \ n \geq1$$ Is this valid?

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  • $\begingroup$ Yes, it looks valid. $\endgroup$ – Simply Beautiful Art Feb 24 '17 at 22:51
  • $\begingroup$ If the pattern will continue forever, yes, the formula is correct. $\endgroup$ – Peter Feb 24 '17 at 22:52
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    $\begingroup$ If you prefer to start your index at $0$, then try $a_n=\left\lceil\frac{n}{2}\right\rceil, \ n \geq 0$. This is OEIS A110654. $\endgroup$ – Alexis Olson Feb 24 '17 at 22:55
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    $\begingroup$ $\dfrac{2n-1+(-1)^n}{4}$ works too. $\endgroup$ – Jonas Meyer Feb 24 '17 at 22:57
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    $\begingroup$ This is of course assuming that the intended pattern is that apart from the term for zero, you have each number appears exactly twice. Technically there is not enough information given to distinguish this from the sequence $\{0,1,1,2,2,3,3,4,4,4,5,5,5,6,6,6,7,7,7,7,8,8,8,8,\dots\}$ where the number of times each number repeats increases as you get further in the sequence, or some other arbitrarily chosen sequence with the same nine initial terms. $\endgroup$ – JMoravitz Feb 24 '17 at 23:13
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In order to make an answer I will write this.

There is a project called OEIS whose purpose is precisely to help people finding closed formulas or at least recognize their sequence in a dictionary of already existing sequences.

So when you have a sequence like yours, you go there, enter your numbers and you will be given possible matches.

As you can notice your formula appears first $a(n)=\lfloor\frac n2\rfloor$ along with its cousin $\lceil\frac n2\rceil$ since it just depends of the starting indice.

But you will notice also that a lot more sequences are matching, why is that ?

This is because 9 terms are insufficient to decide, it could also be the examples below (I picked a few interesting ones). I also indicated where the sequences were branching :

  • $\lfloor n^\frac34\rfloor=0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6,\blacksquare\ 6, 7, 7, 8, 8, 8, 9, 9, 9, 10, ...$

  • decimal expansion of $\frac{10}{891}=0,0112233445566778\blacksquare900112233...$

  • starting from water icing temperature, converting Farenheit to Celsius $\frac59(t-32)=0, 1, 1, 2, 2, 3, 3, 4, 4, 5, \blacksquare\ 6, 6, 7, 7, 8, 8, 9, 9, 10, 11, ...$

  • The minimum number of distinct distances determined by $n$ points in the Euclidean plane (no formula) $0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, \blacksquare\ 5, 6$

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  • $\begingroup$ Thank you very much for the additional information! Very clear! $\endgroup$ – user372003 Feb 25 '17 at 1:47

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