# Equation involving hypergeometric functions

I want to solve this equation but nor NSolve nor Solve are able to do this.

Gamma[1 + 8 (-1 + r) r]/(Gamma[1 + r] Gamma[1 - 9 r + 8 r^2]) - (
Gamma[1 + 14 r] Gamma[
1 + 8 (-2 + r) r] HypergeometricPFQRegularized[{1, -((15 r)/2), -(
r/2)}, {1 + (13 r)/2, 1 - (33 r)/2 + 8 r^2}, 1])/(
Gamma[1 + r/2] Gamma[1 + (15 r)/2])==0

I want to find a solution for $r \in \mathbb{N}$, $r>2$.

NSolve[Gamma[1 + 8 (-1 + r) r]/(
Gamma[1 + r] Gamma[1 - 9 r + 8 r^2]) - (
Gamma[1 + 14 r] Gamma[
1 + 8 (-2 + r) r] HypergeometricPFQRegularized[{1, -((15 r)/2), -(
r/2)}, {1 + (13 r)/2, 1 - (33 r)/2 + 8 r^2}, 1])/(
Gamma[1 + r/2] Gamma[1 + (15 r)/2]), r, Integers]

Any suggestion? I get the following error message:

NSolve::nsmet: This system cannot be solved with the methods available to NSolve.
-

I wonder if this equation has no solution for r > 2 (integer or real). If I create a function of the left-hand side of the equation and plot that against r (at least up to r=50), the relationship of Log[-f[r]] vs. r looks pretty linear and I don't see the function heading back to zero (ever).

f[r_] := Gamma[
1 + 8 (-1 + r) r]/(Gamma[1 + r] Gamma[1 - 9 r + 8 r^2]) - (Gamma[
1 + 14 r] Gamma[
1 + 8 (-2 + r) r] HypergeometricPFQRegularized[{1, -((15 r)/
2), -(r/2)}, {1 + (13 r)/2, 1 - (33 r)/2 + 8 r^2},
1])/(Gamma[1 + r/2] Gamma[1 + (15 r)/2])

ListLinePlot[Table[{r, Log[-f[1.*r]]}, {r, 1, 50}]]

with output

-
For whatever it's worth: The evaluation of your equation happens relatively quickly for even integers but painfully slowly for odd integers. The only way I could get a result for r=49 was to evaluate it with "49.". In any event, the value for your equation gets larger and larger with increasing even integers with no obvious end in site. Is there some reason to suspect the function goes to zero or even decreases at some point? – Jim Baldwin May 31 '15 at 6:12