  The recently developed on-shell bootstrap for computing one-loop amplitudes
in non-supersymmetric theories such as QCD combines the unitarity method with
loop-level on-shell recursion. For generic helicity configurations, the
recursion relations may involve undetermined contributions from non-standard
complex singularities or from large values of the shift parameter. Here we
develop a strategy for sidestepping difficulties through use of pairs of
recursion relations. To illustrate the strategy, we present sets of recursion
relations needed for obtaining n-gluon amplitudes in QCD. We give a recursive
solution for the one-loop n-gluon QCD amplitudes with three or four
color-adjacent gluons of negative helicity and the remaining ones of positive
helicity. We provide an explicit analytic formula for the QCD amplitude
A_{6;1}(1^-,2^-,3^-,4^+,5^+,6^+), as well as numerical results for
A_{7;1}(1^-,2^-,3^-,4^+,5^+,6^+,7^+), A_{8;1}(1^-,2^-,3^-,4^+,5^+,6^+,7^+,8^+),
and A_{8;1}(1^-,2^-,3^-,4^-,5^+,6^+,7^+,8^+). We expect the on-shell bootstrap
approach to have widespread applications to phenomenological studies at
colliders.
