[r] Extract every nth element of a vector

I think you are asking two things which are not necessarily the same

I want to extract every 6th element of the original

You can do this by indexing a sequence:

foo <- 1:120
foo[1:20*6]

I would like to create a vector in which each element is the i+6th element of another vector.

An easy way to do this is to supplement a logical factor with FALSEs until i+6:

foo <- 1:120
i <- 1
foo[1:(i+6)==(i+6)]
[1]   7  14  21  28  35  42  49  56  63  70  77  84  91  98 105 112 119

i <- 10
foo[1:(i+6)==(i+6)]
[1]  16  32  48  64  80  96 112