本文由三個結合 tan−1 與費波那契數列的等式及其所搭配的無字證明圖形出發, 做出和盧卡斯數列有關的圖形, 並由數學歸納法找出並證明 tan−1
與盧卡斯數列及一般二階線性遞迴數列的全新等式:

This paper startes with three equations of tan−1 and the Fibonacci sequence combined
with the diagrams used to prove the three equations without words. According to
the principle of mathematical induction, we continued to find out the similar equations
of the Lucas numbers and the second-order liner recursive sequences as follows.

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