【摘要】 本文探讨通项公式非常相似的斐波那契数列{F_n}和卢卡斯数列{L_n}之间新的关系、性质和变化趋势.发现任何一个卢卡斯数L_n均可表达成两个斐波那契数F_n+1,F_n-1之和,而两个卢卡斯数L_n+1,L_n-1之和却等于5F_n;在讨论{F_n}和{L_n}前后比值数列{a_n/a_n+1}趋近于黄金数时,发现{a_n/a_n+1}的奇偶子列具有严格单调性和有界性;最后给出下一步关于{F_n}和{L_n}的研究思路.更多还原

【Abstract】 Based on the similarity of Fibonacci sequence {F_n} and Lucas sequence {L_n}, this paper explores their new connections, properties and the changing trends. It is found that any Lucas number can be expressed as the sum of two Fibonacci numbers, and the subsequences {a_2n-1/a_2n} and {a_2n/a_2n+1} of the ratio sequences {a_n/a_n+1} of {F_n} and {L_n} are strictly monotonic and bounded and convergent to the golden number. Some applications of {F_n} and {L_n} are also given.更多还原