How many ways are there to tile dominos (with size $2 × 1$) on a grid of $2 × n$?

How about on a grid of $3 × 2n$?

  • $\begingroup$ I am not too familiar with dominos (strange I know) but they are cut half way so label those $A,B$. Here $2x1$ is two rows one column. So I can have A on top of B or flip and put B on top of A so would that be two ways for you? $\endgroup$ – Mr.Fry Jun 9 '15 at 1:51
  • $\begingroup$ oeis.org/A000045, oeis.org/A001835 $\endgroup$ – David Jun 9 '15 at 2:14
  • $\begingroup$ You can find this in Chapter 7.1 of Concrete Mathematics by Graham, Knuth, and Patashnik. $\endgroup$ – user109360 Jun 9 '15 at 11:01

Let $ f_n$ be the number of ways to tile a 2 by n grid with dominoes.

Consider the first square in the grid (the leftmost square in the first row),

A) you could place a domino vertically on that, so there are $ f_{n-1}$ ways to tile the rest of the grid.

B) or you could place the domino horizontally on it, so you must place another domino horizontally below that and there are $ f_{n-2} $ ways to tile the rest of the grid.

So we have $ f_n= f_{n-1} + f_{n-2} $. We need to calculate $f_1$ and $f_2$ and then we can calculate $ f_n$ for any natural number.

It is not hard to see that $f_1=1$ and $f_2=2$.

For the 3 by 2n grid, let $f_n$ be the number of ways to tile the 3 by 2n grid with dominoes and $g_n$ be the number of ways to tile a 3 by 2n+1 grid missing its first square.

Use the above method to show that, $$ f_n=2g_{n-1}+f_{n-1} $$ and $$ g_n=f_n+g_{n-1} $$

Now you should just solve these equations to see that $$ f_{n+1}=4f_n-f_{n-1} $$

And again you should also calculate $f_1$ and $ f_2$.

  • $\begingroup$ +1, but it is often better to start with $f_0$ in problems like this. Here $f_0=1$. Now you are left only to calculate $f_1$. $\endgroup$ – Thomas Andrews Jan 12 '17 at 20:40
  • $\begingroup$ @ Pegah Pournajafi I think $f_2 = 4$ for the $2\times n$ case. The first domino can be placed left vertical, right vertical, top horizontal, or bottom horizontal. $\endgroup$ – user308973 Jan 17 '17 at 23:13
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    $\begingroup$ Shouldn't the formula for $f_n$ be $$f_n = n f_{n-1} + n(n-1) f_{n-2}$$ to account for the fact that we have choices for which dominoes we initially place down? $\endgroup$ – user308973 Jan 17 '17 at 23:31
  • $\begingroup$ @ThomasAndrews Yes, this is much simpler. $\endgroup$ – Pegah Pournajafi Jan 18 '17 at 16:34
  • $\begingroup$ This problem is assuming we have identical dominoes, not different dominos. @user308973 $\endgroup$ – Thomas Andrews Jan 18 '17 at 16:45

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