Hi Pieter,
I guess what you're looking for is
FindSequenceFunction.FindSequenceFunction attempts to find a simple function that yields the sequence a_n.
Mathematica does recognise your first sequence as
Fibonachi[n,2]
FindSequenceFunction[{1, 2, 5, 12, 29, 70, 169, 408}, n]
=> Fibonacci[n, 2]
Which is, according to oeis.org (online encyclopedia of integer sequences) correct, since
Pell-numbers are defined as:
a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2).
Table[Fibonacci[n, 2], {n, 9}]
=> {1, 2, 5, 12, 29, 70, 169, 408, 985}
Your second sequence is recognised by Mathematica as well:
func = FindSequenceFunction[{1, 2, 5, 12, 29, 70, 169, 408, 969}, n]
=> 1/630 (2274 n - 3444 n^2 + 2569 n^3 - 945 n^4 + 196 n^5 - 21 n^6 + n^7)
If we replace now n with a range [1,10]:
func /. n -> {Range[10]}
=> {1, 2, 5, 12, 29, 70, 169, 408, 969, 2218}
If, by any matter, Mathematica does not find a generating function for an integer sequence you might
consult
http://oeis.org. This is normally my first stop when I search for a specific integer sequence.
Hope this helps.
