Fibonacci Rectangles and Triangular Numbers

  • Thread starter ramsey2879
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The sum of the first n Fibonacci Rectangles is given by Sloane's Encyclopedia of Integer Sequences, https://oeis.org/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 https://oeis.org/A000217, 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?
 
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fresh_42

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There are countless formulas and identities for Fibonacci numbers: cp. Catalan, Cassini, d'Ocagne, Lucas.
It is not surprising if you compare twice the area of a triangle with the area of a rectangle that there will be dependences.
 

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