What sequences where the difference between their consecutive terms is always a fibonacci numbers?

Question

What sequence where the difference between its consecutive terms is always a fibonacci numbers ?

I am trying to figure out a pattern in this sequence :

1,2,4,7,12,20,33,54,88

Answer 1

Try adding $1$ to each number and see what you get.

Answer 2

Maybe it is $a(n) = a(n-1)+a(n-2)+1,$ $a(0)=-1,a(1)=1.$ Then $a(2)=2, a(3)=4$ and so on.

Answer 3

We will proceed mechanically. Let $a_n$ be the $n$-th term of our sequence. Let $b_n=a_{n}-a_{n-1}$. We are told that the sequence $(b_n)$ obeys the Fibonacci recurrence, so $b_{n+1}=b_n+b_{n-1}$.

Substituting, we obtain $$a_{n+1}-a_n=a_n-a_{n-1}+a_{n-1}-a_{n-2},$$ which simplifies to $$a_{n+1}=2a_n-a_{n-2}.$$ Now use your favourite method. One way is to note that the characteristic polynomial is $x^3-2x^2+1$.