Question

已知数列{an},如果数列{bn}满足b1=a1,bn=an+a(n-1)则称数列{bn}是数列{an}的生成数列

Answer 1

d(n)=2^n + n,

p(1)=d(1)=2^1 + 1 = 3,

p(n+1)=d(n+1)+d(n)=2^(n+1)+(n+1) + 2^n + n = 3*2^n + 2n+1,

L(2n-1)=d(2n-1)=2^(2n-1)+(2n-1),

L(2n)=p(2n)=p(2n-1+1)=3*2^(2n-1)+2(2n-1)+1,

L(2n-1)+L(2n)=4*2^(2n-1)+3(2n-1)+1=8*4^(n-1) + 6n - 2

T(2n)=L(1)+L(3)+...+L(2n-1)+L(2)+L(4)+...+L(2n)

=8[1+4+...+4^(n-1)] + 6[1+2+...+n] - 2n

=8[4^n - 1] /(4-1) + 6n(n+1)/2 - 2n

=8(4^n - 1)/3 + 3n(n+1) - 2n

T(2n-1)=T(2n)-L(2n)=8(4^n-1)/3 + 3n(n+1)-2n - 3*2^(2n-1) - 2(2n-1)-1

=8(4^n-1)/3 + 3n(n+1)-2n - (3/2)*4^n - 4n + 1

=(7/6)4^n + 3n^2 - 3n - 5/3