It's well known that difference tables can be used to find polynomial functions that go with a sequence of numbers. E.g:(adsbygoogle = window.adsbygoogle || []).push({});

0 1 4 9 16 25 ...

1 3 5 7 9 ...

2 2 2 2 ...

0 0 0 ...

The polynomial can be recovered by taking 0*1 + 1*x + 2*x(x-1)/2! + 0*x(x-1)(x-2)/3! + 0 ... = x + x^2 - x = x^2

If the function is not polynomial, though, the difference table never terminates. I have been looking at when the function is an exponential and have found some curious properties. First, the sequence going down the first diagonal, which would normally be used to reconstruct the polynomial, is also an exponential, with the base reduced by 1.

1 3 9 27 81 243 ...

2 6 18 54 162 ...

4 12 36 108 ...

8 24 72 ...

16 48 ...

32 ...

This is easily proven in general. I think I've found a method of taking a sequence and iteratively make difference tables for it to determine its function. If the function is a sum of exponentials, all with whole number bases, it's fairly easy. The product of a polynomial with an exponential is a bit less trivial, but still fairly easy to do. I've figured out how to do a few special cases of a sum of these products, but I'm having trouble generalizing those special cases. For instance, the sequence 0, 7, 68, 435, 2320, 11195, 50820, 221851, ... can be found to be x^2 * 3^x + x * 4^x, but I'm having a terrible time getting that 1, 6, 32, 136, 512, 1824, 6400, 22656, 81920, ... is x^2 * 2^x + 4^x from the difference tables.

I'm not sure whether what I'm doing is novel, so if any of you know of something like this being done before, I'd love to find out. And if any of you are interested in the methods I've used to determine the functions, I'll be happy to expound.

Btw, sorry about the formatting - I haven't had the chance to figure out how to make this look pretty yet.

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# Difference tables

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