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trigonometry - What is the connection and the difference between the . . . Around 1200, mathematician Leonardo Fibonacci discovered the unique properties of the Fibonacci sequence This sequence ties directly into the Golden ratio because if you take any two successive Fibonacci numbers, their ratio is very close to the Golden ratio As the numbers get higher, the ratio becomes even closer to 1 618
geometry - Where is the pentagon in the Fibonacci sequence . . . The Fibonacci sequence is related to, but not equal to the golden ratio There is no reason to expect that the sequence mimics the geometric series $\varphi^n$ than there is to expect that the Fibonacci spiral is the same as the golden spiral
What is the meaning of limit of Fibonacci sequence? The existence of the limit reflects the fact that the Fibonacci sequence is essentially a geometric sequence (it is actually a linear combination of two geometric sequences but one of them dominates the other)
recurrence relations - Fibonacci, tribonacci and other similar . . . Whoever invented "tribonacci" must have deliberately ignored the etymology of Fibonacci's name - which was bestowed on him quite a bit after his death Leonardo da Pisa's grandfather had the name Bonaccio (the benevolent), which was also used by his father The name "filius bonacii" or "figlio di Bonaccio" (son of Bonaccio) was contracted to give Fibonacci By the way: the Fibonacci sequence
Strong Induction Proof: Fibonacci number even if and only if 3 divides . . . 0 Since the period of $2$ in base $\phi^2$ is three places long = $0 10\phi\; 10\phi \dots$, and the fibonacci numbers represent the repunits of base $\phi^2$, then it follows that $2$ divides every third fibonacci number, in the same way that $37$ divides every third repunit in decimal (ie $111$, $111111$, $111111111$, etc)