# 피보나치 수열의 짝수항

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## 개요

• 피보나치 수열 $\{F_{n}\}_{n\geq 0}$의 짝수항으로 주어지는 수열, 즉 $\{F_{2n}\}_{n\geq 0}$

$$1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181,\cdots$$

• 점화식 $a_{n+2}=3a_{n+1}-a_{n}$
• 다음을 만족한다

$$a_{n+2}=\frac{1+a_{n+1}^2}{a_{n}}$$

• 불변량

$$\frac{a_{n-1}^2+a_{n}^2+1}{a_{n-1}a_{n}}=3$$

## 일반적인 초기조건

• 점화식

$$a_{n+2}=\frac{1+a_{n+1}^2}{a_{n}}, a_{0}=\alpha, a_{1}=\beta$$

• 다음의 선형점화식을 만족한다

$$a_{n+2}=\frac{1+\alpha ^2+\beta ^2}{\alpha \beta }a_{n+1}-a_{n}$$

• 처음 몇 개의 항은 다음과 같다

$$\alpha ,\beta ,\frac{1+\beta ^2}{\alpha },\frac{\alpha ^2+\left(1+\beta ^2\right)^2}{\alpha ^2 \beta },\frac{\alpha ^4+2 \alpha ^2 \left(1+\beta ^2\right)+\left(1+\beta ^2\right)^3}{\alpha ^3 \beta ^2},\frac{\alpha ^6+3 \alpha ^2 \left(1+\beta ^2\right)^2+\left(1+\beta ^2\right)^4+\alpha ^4 \left(3+2 \beta ^2\right)}{\alpha ^4 \beta ^3}$$

• 불변량

$$\frac{a_{n-1}^2+a_{n}^2+1}{a_{n-1}a_{n}}=\frac{\alpha ^2+\beta ^2+1}{\alpha \beta }$$

## 관련논문

• Alperin, Roger C. 2011. “Integer Sequences Generated by $x_{n+1}=\frac {x^2_n+A}{x_{n-1}}$.” The Fibonacci Quarterly. The Official Journal of the Fibonacci Association 49 (4): 362–365. http://www.math.sjsu.edu/~alperin/IntegerA-Sequences.pdf
• Caldero, Philippe, and Andrei Zelevinsky. 2006. “Laurent Expansions in Cluster Algebras via Quiver Representations.” Moscow Mathematical Journal 6 (3): 411–429, 587.