May4-08, 11:10 PM
1. The problem statement, all variables and given/known data
Find a linear homogeneous recurrence relation satisfied by an=2^n+n!
2. Relevant equations
3. The attempt at a solution
The teacher gave us a hint using generating functions. The generating function for f(x) is
f(x)=1+2x+4x^2+8x^3+....+1+x+2x^2+6x^3+24x^4+... The first part is a geometric series which equals 1/(1-2x), so f(x)=1/(1-2x)+1+x+2x^2+6x^3+24x^4+... Then he said to multiply both sides by x, take the derivative, and relate f'(x) to f(x). I found f'(x) after multiplying by x to be f(x)+xf'(x)=1/((1-2x)^2)+1+2x+6x^2+24x^3+... Now I don't know how to proceed. I need to relate the two, but am lost as to how to do that. Any suggestions? Thanks.
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