一类三角形数转化成正方形数的充要条件及算法研究  

Study on the necessary and sufficient condition and algorithm for a class of triangle numbers to be transformed into square numbers

作　　者：赵美利[1] 唐静[1] ZHAO Meili;TANG Jing(Chuzhou City Vocational College,Chuzhou 239000,China)

机构地区：[1]滁州城市职业学院,安徽滁州239000

出　　处：《玉溪师范学院学报》2022年第3期21-29,共9页Journal of Yuxi Normal University

基　　金：安徽省职业与成人教育一般项目“区域民俗旅游经济发展与高职教育良性互动的策略研究——基于‘大江北’规划”(项目编号:Azcj104);安徽省职业与成人教育学会2019年度教育科研规划课题“5G时代高职计算机应用混合式教学改革探究”(课题编号:Azcj118);滁州城市职业学院人文科研重点项目“大数据专业学生数学建模素养的影响因素分析和研究”(项目编号:2020sk09);2021年校级质量工程“课程思政会计教学团队”项目(项目编号:2021szjxtd04).

摘　　要：结合数列性质和佩尔方程,讨论了三角形数转化成正方形数的充要条件.设此三角形数的第n行的正整数为a_(n),把a_(n)从小到大构成一数列{a_(n)},则数列{a_(n)}是无穷数列,{a_(n)}有递推公式a_(n+2)=6a_(n+1)-a_(n+2),a_(1)=1,a_(2)=8,其通项公式为a_(n)=1/4[(3+2√2)^(n)+(3-2√2)^(n)]-1/2,且三角形数可化为正方形数.对于该结论,利用Java语言的大整数特征结合Eclipse集成开发环境对其进行验证,证明了结论的正确性.The necessary and sufficient conditions for transformation of triangle number into the square number are discussed based on the nature of the sequence and Pell’s equation.Let the positive integer at the n row of the triangle number be a_(n).If a_(n) form a sequence{a_(n)}in a_(n) ascending order,then the{a_(n)}sequence is infinite and has a recurrence formula a_(n+2)=6a_(n+1)-a_(n+2),a_(1)=1,a_(2)=8 with a general formula of a_(n)=1/4[(3+2√2)^(n)+(3-2√2)^(n)]-1/2,and the triangular numbers can be changed into the square number.The conclusion is verified by combining the feature of large integer of JAVA language with Eclipse integrated development environment.

关 键 词：三角形数 正方形数 佩尔方程 大整数 

分 类 号：O156.4[理学—数学]