analytical expression for n^th term can be found by generating polynomial which I learned from Hari sir.

We have to find :    such that 

Adding these three :

except 1st terms all terms are 0 since they follow a_n+2 = 3a_n+1 + 2a_n

we know a₁ = 1 and a₂ = 1. Expand RHS and compare coefficients of xⁿ to find out the analytical expression for a_n ( feeling a bit lazy so didn't do till last step :D )

however I can't figure out what will be the sum of the series