Fibonacci Rectangles and Triangular Numbers

1. Dec 19, 2006

ramsey2879

The sum of the first n Fibonacci Rectangles is given by Sloane's Encyclopedia of Integer Sequences, http:www.research.att.com/~njas/sequences/A064831 .
There the sequence is given as a_n = {0,1,3,9,24,64, ...}

If 2*T(a_n) = a_n*(a_n +1) where T(n) is the nth triangular number, a interesting and new observation is that 2*T(a_n) = F_(n-1)*F_n*F_n*F_(n+1). Thus a_n = the integer part of the square root of the product of adjacent Fibonacci Rectangles.

Can anyone explain why triangular numbers are so closely tied to Fibonacci numbers?