Let me show you how to make that awk solution more idiomatic (refer to the awk info page on stackoverflow).

Starting with:

$ seq 10 20 | awk '{if ($0 ~ /11/) {n=NR+6}; if (NR<n) next; print $0}'

10

17

18

19

20

First, we'll take the if statements, and turn them into condition {action} lines

awk '

    $0 ~ /11/ {n=NR+6}

    NR < n {next}

    {print $0}

'

Then, we'll use $0 as the default value for some things:

awk '

    /11/ {n=NR+6}

    NR < n {next}

    {print}

'

Then, we'll take {print} as the default action if the condition is true

awk '

    /11/ {n=NR+6}

    NR < n {next}

    1

'

Then, we'll invert the NR < n condition to remove the next

awk '

    /11/ {n=NR+6}

    NR >= n

'

And we can one-liner-ize it:

awk '/11/ {n=NR+6} NR >= n'

This produces the same output

$ seq 10 20 | awk '/11/ {n=NR+6} NR >= n'

10

17

18

19

20

Comparing:

awk '{if ($0 ~ /11/) {n=NR+6}; if (NR<n) next; print $0}'

awk '/11/ {n=NR+6} NR >= n'

As a last step, you might want to pass the pattern and the value of N as parameters into awk, so they don't have to be hardcoded. This inflates the awk call but might be more flexible for you:

awk -v patt="11" -v N=6 '$0 ~ patt {n = NR + N} NR >= n'

By placing the parameters after the awk script, -v can be dropped, making the command a bit shorter:

awk '$0 ~ patt {n = NR + N} NR >= n' patt=11 N=6

One advantage to putting the variables after the awk body is they can get different values for each file:

awk '$0 ~ patt {n = NR + N} NR >= n' patt=11 N=6 file1 N=10 file2