Interesting Series

Warning - (Simple) Math Post

So I came across a series today and had to guess the next term. I did, I think correctly, but in doing so found two different ways of doing so that are fairly different.

Here's the series: 1, 4, 10, 19, 31, ?

It's kind of embarrasing, but the first solution I found was the overly complicated one (it at least matches the next solution on all terms that I checked); writing out 3 lines of numbers makes it easier to see..

01 04 10 19 31 ??
--------------------
01 02 03 04 05 06 <--square this row
00 00 01 03 06 10 <--add this row to the one above
--------------------
00 01 02 03 04 05 <--the amount by which the row above this increments

So, to get the next term, square 6 and add 10 to it to get 46. Each term is a sum of a square and a member of some common increasing series.

The simple way to move from one term to the next is to add the next multiple of 3. Or, to just say that a_n = 3(n-1) + a_n-1, where n is an integer in (2,∞) and a₁=1.

01 04 10 19 31 ??
--------------------
03 06 09 12 15 <--the amount by which the row above this increments

So neither of these is in closed form. The second is the procedure I should've come up with first - it's much simpler. Perhaps later I'll try to see if I can get this to fit into a nice non-recursive formula. Or maybe one of you math/CS smart readers can save me the trouble. ;)