A049982 is the number of arithmetic progressions of 2 or more positive integers, strictly increasing with sum n.

When I stumbled on this sequence (and it's brothers and sisters, with various slightly different qualifications), I noticed a complete lack of any formulas or generating functions that help understand the sequence.  So I did some amateur investigation on my own.

I started by considering the number of arithmetic progressions of 2 positive integers, strictly increasing with sum n.  By convention, I like to start with n=0, so this sequence is 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, ... which has generating function x³/(x³-x²-x+1) which I will rewrite as x³/(x³-x-x²+1) for reasons that will become clear later.

The sum of an arithmetic progression of 3 positive integers is always three times its middle term, hence a multiple of 3.  This sequence is 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 3, 0, 0, 4, ... which has generating function x⁶/(x⁶-2x³+1), which I will write as x⁶/(x⁶-x³-x³+1) for reasons that will become clear later.

The sum of an arithmetic progression of 4 positive integers follows a pattern that's a little harder to discern.  But if I give you enough terms, I think you can start to pick up the rhythm:  0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 2, 0, 2, 0, 2, 0, 3, 0, 2, 0, 3, 0, 3, 0, 3, 0, 3, 0, 4, 0, 3, ...  This has generating function x¹⁰/(x¹⁰-x⁶-x⁴+1)

As you can imagine, I kept going.  As the going got tougher, I started inventing little tools to help, such as PuzzleGeneratingFunction.xls, an Excel spreadsheet that guesses the generating function for a given series. (. . . . . . maybe I'll write a page about that spreadsheet some day.)

After a while, I had a little table that shows the

generating function for the sequence of the number of arithmetic progressions of k positive integers, strictly increasing with sum n:

Now, maybe you can see why I wrote the terms for k=2 and k=3 in such a funny way.  In general, the generating function for the sequence of the number of arithmetic progressions of k positive integers, strictly increasing with sum n is:

x^t(k)/(x^t(k)-x^t(k-1)-x^k+1), where t(k) is the k'th triangular number

Summary

A049982 has generating function x³/(x³-x-x²+1) + x⁶/(x⁶-x³-x³+1) + x¹⁰/(x¹⁰-x⁶-x⁴+1) + ... which is the
sum k=2,3,... of x^t(k)/(x^t(k)-x^t(k-1)-x^k+1), where t(k) is the k'th triangular number
Term k of this generating function generates the number of arithmetic progressions of k positive integers, strictly increasing with sum n.

Internet References

A049982 -- The Online Encyclopedia of Integer Sequences.

Related pages in this website

See also Recurrence Relation

The webmaster and author of the Math Help site is Graeme McRae.
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