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I am trying to find a formula to calculate the following sum: $$\sum_{x=0}^n (2ax - {1 \over 2}a^2 - {1 \over 2} a) $$

where

$$ a = \left\lfloor {x \over \phi^2} \right\rfloor $$

and

$$\phi = {1 + \sqrt 5 \over 2}$$

$\checkmark$

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