On the 1st of May, 1795, two new stars were added to the flag of the United States to represent the new states of Vermont and Kentucky, bringing the total up to 15. Two stripes were also added, which made sense at the time, but as more states joined the union over the years, the flag remained unchanged for fear that it would become too complex. It was waith the Flag Act of 1818 that the current standard of 13 stripes and a star for every state was established in law.


Some amounts of stars allow for rather nice looking flags, while others do not. For the past century, only a handful of new states have been added, so it's fortunate that 48, 49, and 50 happen to be numbers that allow for nice flags, but what if a new state is added?
There have been talks of Puerto Rico or the District of Columbia becoming states, and there are several other territories that could theoretically become states some day, so we have to ask: How long will the US flag be safe from the horrors of misaligned stars?

The historical flags that look the nicest generally fall into one of two types: Stars aligned to a rectangular lattice, like in the 48-star flag, and those aligned to a rhombic lattice, like the 50-star flag.


The proportion of the whole flag is $10:19$, but as the canton extends $2/5$ of the width and covers $7/13$ stripes, the canton's proportion is $175:247$, very close to $1:\sqrt{2}$. Based on the historical flags with rectangular star patterns, precedent exists for using an $n\times m$ block of stars if $1<m/n<2$. This restriction allows for about a quarter of the natural numbers to be represented.[1]A001284, conjectured to contain 1/4 of the natural numbers.
Star patterns aligned to a rhombic lattice correspond to half of a rectangular lattice. The 50-star flag uses every other star within what would be a $9\times 11$ grid, and by using the other stars, a 49-star block could be made. In general, in an $n\times m$ grid, if $n*m$ is odd, either $(n*m+1)/2$ or $(n*m-1)/2$ star patterns can be found, while if $n*m$ is even, there is only a $(n*m)/2$ star pattern. Rhombic lattice can bend more without becoming cramped, and by scaling in one direction by a factor of $\sqrt{3}$, a regular rhombic lattice becomes triangular, so I used the alternating patterns in any $n\times m$ block of stars such that $1/\sqrt{3}<m/n<2\sqrt{3}$.
These sets of numbers seem to allow for about three quarters of the natural numbers to be represented, though determining the exact amount is beyond my skill with number theory. What I can say for certain however, is that any star count from 51 to 61 can be represented.


With the 62nd state, either someone more creative than I will have to come up with a new pattern, or we'll be stuck with the kind of imperfection that so many of the United States' historical flags had. However, I wouldn't plan on this becoming a problem any time soon.