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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?

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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