In the era of internet, you can find specialised search engines for many things. Sequences aren't an exception. There are 250,000 sequences in OEIS database.
Let's try to find some interesting elementary sequences containing 4,8,12.
The most obvious sequence is Multiples of 4. Then, there is Fibonacci sequence beginning 0, 4. Less trivial are Product of decimal digits of n with n=41,42,43 (though its next two numbers are no different from the arithmetic progression) or Alternately add and multiply.
I explained first three to my daughter but she liked her answer (the first one) more. Then she got bored and switched to reading a book.
For older children, there are nice sequences Floor(n^2/2), Numbers n such that 9*n+1 is prime or my favourite The middle member 'b' of the Pythagorean triples (a,b,c) ordered by increasing c (and similar sequences A009012 and A009023).
It's time to stop giving sequences for homework because smart kids with internet can easily make their teacher feel dumb. Or even bring the teacher to tears with Number of divisors of Catalan number A000108(n).