# Recursion Puzzle

Leroy Quet qqquet at mindspring.com
Tue Sep 3 04:14:30 CEST 2002

```I posted this recursion puzzle on sci.math and rec.puzzles a couple of
days ago. I was debating with myself whether to post it to this group,
however. But I decided to go ahead and indeed post it.
(Perhaps it will be fun for some of you to try and solve it.)

---

The following recursion generates a sequence of positive integers. (The
sequence may or may not be in the EIS.) -- The fact that the sequence
contains all rationals, let alone positive integers, would not be obvious
to ME just from looking at the recursion.

Let a(0) = 1;
a(1) = 2;
a(2) = 5;

and, for m >= 3,

a(m) =

(1/4)*( (3*a(m-2)^2 *b(m) -9*a(m-2)^3)/a(m-3)^2

+(13*a(m-1)*a(m-2) -8*a(m-2)^2 +2*a(m-2)*b(m) -3*a(m-1)b(m))/a(m-3)

+8*a(m-1) -2*a(m-2) ),

where b(m) =

sqrt(9*a(m-2)^2 +4*a(m-3)*(a(m-2) -2*a(m-1))).

Find a closed form, in terms of a partial sum, for the sequence of a(m)'s.

(I'll give my answer in a few days if no one else gets it sooner.)

Thanks,
Leroy Quet

```