## how to derive n[k]

Hello,
I am a physics student and had an interesting problem that I have solved, but I don't know how to proceed from here, I have some things mixed up.

I have derived the following condition:

n[i]=3*n[i+1]+7*n[i+2]+11*n[i+3]+15*n[i+4]+19*n[i+5]+23*n[i+6] +... etc. until maximum i.

for all n[i] this condition has to be satisfied.

I want to have a function that gives n[i] given the maximum i. So say I know that i-max is 10 I want to be able to derive all n[i].

The only thing I could think of was to start from maximum i and give it value 1 and then calculate (maximum i - 1) then (maximum i - 2) etcetera. The number you substract from the maximum i I will denote with k. The result is the same for each maximum i:

k...n[k]...3n[k-1]..7n[k-2]..11n[k-3]..15n[k-4]..19n[k-5]..23n[k-6]
0...0
1...3.......3
2...16......9..........7
3...60......48........21........11
4...340....180.......112......33.............15
5...1680..1020......420......176...........45..... .....19
6...8400..5040......2380....660...........240..... ....57..........23

I tried just to find n[k] as a function of k, but I didn't succeed.

Could anyone help me find the relation of n[k] as a function of k, or even beter a function of n[i] given a certain maximum i.

Thanks,

Djehuti