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 Math Help > Sequences and Series > Sequences > A049982

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:

 k Generating Function 2 x3/(x3-x-x2+1) 3 x6/(x6-x3-x3+1) 4 x10/(x10-x6-x4+1) 5 x15/(x15-x10-x5+1) 6 x21/(x21-x15-x6+1) 7 x28/(x28-x21-x7+1) 8 x36/(x36-x28-x8+1) 9 x45/(x45-x36-x9+1) 10 x55/(x55-x45-x10+1) 11 x66/(x66-x55-x11+1)

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:

xt(k)/(xt(k)-xt(k-1)-xk+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 xt(k)/(xt(k)-xt(k-1)-xk+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.