functions together with factorials and multinomial coefficients then it appears that there is not. But if one also allows higher mathematical functions then it turns out that such a formula can in fact be found: as indicated in the table above each coefficient is given by a particular value of a so-called Gegenbauer or ultraspherical function.
So what about other nested patterns? Both of the patterns shown on the previous page are rather special in that as well as being generated by substitution systems they can also be produced one row at a time by the evolution of one-dimensional cellular automata with simple additive rules. And in fact the approaches used above can be viewed as direct generalizations of such additive rules to the domain of ordinary numbers.
For a few other nested patterns there exist fairly simple connections with additive cellular automata and similar systems—though usually in more dimensions or with more neighbors. But for most nested patterns there seems to be no obvious way to relate them to ordinary mathematical functions. Nevertheless, despite this, it is my guess that in the end it will in fact turn out to be possible to get a formula for any nested pattern in terms of suitably generalized hypergeometric functions, or perhaps other functions that are direct generalizations of ones used in traditional mathematics.
Yet given how simple and regular nested patterns tend to look it may come as something of a surprise that it should be so difficult to represent them as traditional mathematical formulas. And certainly if this example is anything to go by, it begins to seem unlikely that the more complex kinds of patterns that we have seen so many times in this book could ever realistically be represented by such formulas.
But it turns out that there are at least some cases where traditional mathematical formulas can be found even though to the eye or with respect to other methods of perception and analysis a pattern may seem highly complex.
The picture at the top of the facing page is one example. A pattern is built up by superimposing a sequence of repetitive grids, and to the eye this pattern seems highly complex. But in fact there is a simple formula for the color of each square: given the largest factor in common between the