Context Free Grammar for language
Production for $\{b^jc^k\; |\; j\neq 2k \}$ can be written as 

$B\rightarrow bBcc\;|\;bB_1\;|\;cB_2$

$B_1\rightarrow bB_1\;|\;b\;|\;c\;|\;\epsilon$

$B_2\rightarrow cB_2\;|\;c\;|\;\epsilon$

You can see that it can't accept $bbcccc$. We can use $B\rightarrow bBcc$ twice but the final $B$ would have to be substituted with either b or c.

It accepts $bbccc$ as $B\rightarrow (bBcc) \rightarrow b(bB_1)cc \rightarrow bb(c)cc$

For even number of cs $B\rightarrow bBcc$ can be used. For odd number of cs, an extra $B_1\rightarrow c$ can be used.

So $\{a^ib^jc^k\; |\; j\neq 2k \}$ has the following grammar

$S\rightarrow AB$

$A\rightarrow aA \;|\; \epsilon$

$B\rightarrow bBcc\;|\;bB_1\;|\;cB_2$

$B_1\rightarrow bB_1\;|\;b\;|\;c\;|\;\epsilon$

$B_2\rightarrow cB_2\;|\;c\;|\;\epsilon$

Note: i can be 0 but j and k can not be 0 simultaneously.