Fibonacci Sums

The Fibonacci series is: F₁=1, F₂=2, for n=2,3,... F_n+1= F_n + F_n-1. The (n+1)-st number is the sum of n-th and (n-1)-th number in the sequence.

Fibonacci numbers have the property that: the sum of the first n numbers of a sequence is contained in the sequence. Do you know of others?

F₁+F₂+F₃+...+F_n = F_n+2-1

Actually the sequence G₁=1, G₂=2,for n=2,3,... G_n+1= G_n + ... + G₂ + G₁ trivially staifies that property. So, Fibonacci sequence is not unique in the above sense. Can you think of a sequence {H_k} such that

H₁+H₂+H₃+...+H_n = H_n+3 - c? for some constant c.

On a tangent: Kolmogorov information complexity speaks of representations that can compress information effectively. Not only does the following set of characters "F₁=1, F₂=2 F_n+1= F_n + F_n-1" contain the entire Fibonacci sequence, but also the sum of its first n elements.