I have solved http://projecteuler.net/problem=44 by getting the inverse equation from Wikipedia http://en.wikipedia.org/wiki/Pentagonal_number:
Pentagonal: $f(n) = \frac{n(3n - 1)}{2}$
Inverse Pentagonal: $n = \frac{\sqrt{24f(n) + 1}+1}{6}$
am interested in the steps from Pentagonal equation (quadratic?) to the Inverse.
I note that it is similar to What is the inverse of $f(n)=\frac{n^2+n}{2}$? and I've tried the same strategy:
$f(n) = \frac{n(3n - 1)}{2}$
*6 + 1 on each side
$6f(n) + 1 = 9n^2 -3n + 1$
but this isn't correct because I want:
$6f(n) + 1 = 9n^2 -6n + 1$
to give:
$(3n-1)^2$ on the right hand side