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We construct the following sequence: Beginning at 0, we move to the next whole number by changing our first number by one, our second number by two, our third number by three, etc. If we are able to move backwards to a whole number that has not been previously listed, we do so, otherwise we move forward. The first five numbers of this sequence are 0, 1, 3, 6, 2. Find the 20th number in this sequence. What is the first number to appear twice in this sequence?

Jun 7, 2020

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We construct the following sequence:
Beginning at 0, we move to the next whole number by changing our first number by one,

our second number by two, our third number by three, etc.
If we are able to move backwards to a whole number that has not been previously listed, we do so, otherwise we move forward.
The first five numbers of this sequence are 0, 1, 3, 6, 2.
Find the 20th number in this sequence.
What is the first number to appear twice in this sequence?

Recaman's sequence:

$$\begin{array}{|r|r|} \hline 1 & 0, \\ & & +1 \\ 2 & 1, \\ & & +2 \\ 3& 3, \\ & & +3 \\ 4& 6, \\ & & -4 \\ 5& 2, \\ & & +5 \\ 6& 7, \\ & & +6 \\ 7& 13, \\ & & +7 \\ 8& 20, \\ & & -8 \\ 9& 12, \\ & & +9 \\ 10& 21, \\ & & -10 \\ 11& 11, \\ & & +11 \\ 12& 22, \\ & & -12 \\ 13& 10, \\ & & +13 \\ 14& 23, \\ & & -14 \\ 15& 9, \\ & & +15 \\ 16& 24, \\ & & -16 \\ 17& 8, \\ & & +17 \\ 18& 25, \\ & & +18 \\ 19& 43, \\ & & +19 \\ \color{red}20& \color{red}62, \\ & & -20 \\ 21& \color{green} 42, \\ & & +21 \\ 22& 63, \\ & & -22 \\ 23& 41, \\ & & -23 \\ 24& 18, \\ & & +24 \\ 25& \color{green} 42, \\ & & -25 \\ 26& 17, \\ & &+26 \\ 27& 43, \\ \ldots & \ldots \\ \hline \end{array}$$

Jun 8, 2020