피보나치 수열의 짝수항

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

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

매스매티카 파일 및 계산 리소스

• https://docs.google.com/file/d/0B8XXo8Tve1cxUl9YZnh0V0pOLTQ/edit?usp=drivesdk
• http://oeis.org/A001519

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.