Question

a(1)=1,a(2)=3,a(n)=a(n-1)+a(n-2)+1,求a(n)

Answer 1

an=a+a+1,①

∴a=an+a+1,②

②-①,a-an=an-a,

∴a-2an+a=0,

特征方程是x^3-2x^2+1=0,解得x1=1,x2=(1+√5)/2,x3=(1-√5)/2.

a1=1,a2=3,a3=5,

设an=x+y[(1+√5)/2]^n+z[(1-√5)/2]^n,则

1=x+[(1+√5)/2]y+[(1-√5)/2]z,

3=x+[(3+√5)/2]y+[(3-√5)/2]z,

5=x+(2+√5)y+(2-√5)z.

解得x=-1,y=(5+√5)/5,z=(5-√5)/5,

∴an=-1+[(5+√5)/5][(1+√5)/2]^n+[(5-√5)/5][(1-√5)/2]^n.