Thursday, March 15, 2007

Fibonacci Sums

The Fibonacci series is: F1=1, F2=2, for n=2,3,... Fn+1= Fn + Fn-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+...+Fn = Fn+2-1

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

H1+H2+H3+...+Hn = Hn+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 Fn+1= Fn + Fn-1" contain the entire Fibonacci sequence, but also the sum of its first n elements.

No comments: