Finding the general term of this sequence

Question

I have been challenged to find the general term or the recurrence relation for this sequence

5,18,44,96,195.... The only hint I have is that this sequence is an applied Fibonacci sequence. Can someone please suggest ways to find the recurrence or the nth term. I have looked at OEIS but found no note this for this particular integer sequence. I have searched at numerous locations but have had no success. Also ,I think the terms of this sequence can be determined in logarithmic time. Any help will be highly appreciated.

Answer 1

A closed form of the sequence is

a(n) = (n*Luc(n+6) - 4*Fib(n))/5

where Luc(k) is the k-th Lucas number, and Fib(k) the k-th Fibonacci number. Since Luc(k) = Fib(k+1) + Fib(k-1), the formula can easily be rewritten to use only Fibonacci numbers.

Answer 2

Can you provide more members of the sequence?

Nvm, I got it.

a[n] = a[n-1] + a[n-2] + Fibonacci[n+5]

Answer 3

Searching for part of sequence 5,18,44 on OEIS, gives A037140. That sequence is of form:

a(n) = F(n+9)-(29+8*n)

Trying similar approach with parts of Fibonacci sequence on F(n+k)-your_sequence(n) gives:

F(n+10)-your_sequence(n) = 50, 71, 100, 137, 182

That sequence has nice progression: 29+21*n+8*n*(n-1)/2. This sequence has presentation:

a(n) = F(n+10)-(29+21*n+4*n*(n-1)), n=0, 1, ...