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While a one-dimensional sequence is represented by a list, a two-dimensional sequence (a sequence of sequences) can be represented by a matrix. Find the pattern behind the following sequence, where a section of the matrix with unknown coordinates is shown below.

$\begin{bmatrix} &\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ \dots& 1 & 5 & 3 & 7 & 1 & 9 & 5 & 13 & 3 &\dots\\ \dots& 4 & 7 & 2 & 5 & 8 & 1 & 10 & 19 & 4 &\dots\\ \dots& 1 & 5 & 9 & 13 & 2 & 6 & 10 & 14 & 3 &\dots\\ \dots& 4 & 1 & 6 & 11 & 16 & 21 & 2 & 7 & 12 &\dots\\ &\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ \end{bmatrix}$

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The pattern is that the number in the $m$th row and $n$th column is

the number $n$, written in base $m$, reversed, and converted back into base 10. The section shown is the 2nd through 5th lines, 4th through 12th columns.

For example, the $4$th row, $11$th column would be calculated like this: $11_{10}=23_4\rightarrow32_4=14_{10}$

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    $\begingroup$ How did you ever figure that out?! $\endgroup$
    – GentlePurpleRain
    Commented Feb 14, 2016 at 13:13
  • $\begingroup$ @GentlePurpleRain Noticing that the second row does +3s, the third row does +4s, and the fourth row does +5s, then converting into the corresponding bases. $\endgroup$
    – f''
    Commented Feb 14, 2016 at 17:12

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