To make the arrangements of 3 dominoes take all the arrangments of 1 domino and add 2 extra dominoes on the right and take all the arrangements of 2 dominoes and add 1 extra domino on the right.

To make the arrangements for 12 dominoes use the arrangements for 10 dominoes and for 11 dominoes in the same way. To find the number of arrangements for 12 dominoes add up the number of arrangements for 10 dominoes and the number of arrangements for 11 dominoes.

In order to find this number you need to find the sequence of numbers of arrangements for 1 up to 11 dominoes which is:

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144

The number of arrangements for 12 dominoes is 233.

To find the number of arrangements of 12 dominoes we have used the rule that generates the Fibonacci sequence. This rule is that each number is the sum of the two numbers before it. We can write down the rule as

A₁₂=A₁₀+A₁₁

where A₁₂ is the number of arrangements of 12 dominoes, A₁₀ is the number of arrangements of 10 dominoes and A₁₁ is the number of arrangements of 11 dominoes. We need here that A₁=1 and A₂=2.

In general, to make all the arrangements for n dominoes, take all the arrangements for n-2 dominoes and place two dominoes on the right hand end, this way round = like the green dominoes in the illustration, and take all the arrangements for n-1 dominoes and place one domino on the right, this way round |, like the single green domino in the illustration. Thus the formula for the number of arangements for n dominoes is given by

A_n = A_n-2 + A_n-1.

This explains why the pattern will continue indefinitely.