Because typing is not very convenient, so let's talk about the train of thought:

Add a (n+1) to the left and right of1to make bn = an+a (n-1);

Then b(n+ 1)=2b(n) and the prime minister is 3, and BN = 3 * 2 (n-1) can be obtained; Therefore, a (n+1)+an = 3 * 2n = a (n+2)-an;

Therefore, it can be concluded that n=0, A0 =1; n= 1，a 1 = 2； N & gt=3 and n is odd, an = 3 * 2 (n-1)-4; N & gt=2 and n is even, an = 3 * 2 (n-1)-2;

Therefore, a long generating function is the sum of f (x) = ar * x r (r from 0 to positive infinity);

The exponential generating function means f (x) = (ar * x r)/r! (r from 0 to positive infinity);

Comparing Taylor expansion of cos3x at 0 with the above formula, we can get the result:

When n is odd, an = 0；; When n is an even number; an=(- 1)^(n/2)* 3^n * x^n / n！