论文标题
复发关系的互惠总和
Sums of Reciprocals of Recurrence Relations
论文作者
论文摘要
There is a growing literature on sums of reciprocals of polynomial functions of recurrence relations with constant coefficients and fixed depth, such as Fibonacci and Tribonacci numbers, products of such numbers, and balancing numbers (numbers $n$ such that the sum of the integers less than $n$ equals the sum of the $r$ integers immediately after, for some $r$ which is called the balancer of $n$; If $n$ is总结中包括，我们有so爵的数字，$ r $称为$ n $的爵士乐人）。我们将先前的工作概括为具有任意系数和TribonAcci数字的两个复发序列的互相总和，并证明我们的方法提供了一些现有结果的替代证明。 我们定义$（a，b）$平衡和sto式数字，其中$ a $ a和$ b $是分别乘以左侧和右侧的常数，并得出描述这些序列的复发关系。我们表明，对于平衡数字，系数$（3,1）$是唯一的，因此每个整数都是一个$（3,1）$平衡数字，并且证明没有一组类似的sto式数字系数。我们还发现了某些没有平衡或sto谐数的系数的模式。
There is a growing literature on sums of reciprocals of polynomial functions of recurrence relations with constant coefficients and fixed depth, such as Fibonacci and Tribonacci numbers, products of such numbers, and balancing numbers (numbers $n$ such that the sum of the integers less than $n$ equals the sum of the $r$ integers immediately after, for some $r$ which is called the balancer of $n$; If $n$ is included in the summation, we have the cobalancing numbers, and $r$ is called the cobalancer of $n$). We generalize previous work to reciprocal sums of depth two recurrence sequences with arbitrary coefficients and the Tribonacci numbers, and show our method provides an alternative proof of some existing results. We define $(a,b)$ balancing and cobalancing numbers, where $a$ and $b$ are constants that multiply the left-hand side and right-hand side respectively, and derive recurrence relations describing these sequences. We show that for balancing numbers, the coefficients $(3,1)$ is unique such that every integer is a $(3,1)$ balancing number, and proved there does not exist an analogous set of coefficients for cobalancing numbers. We also found patterns for certain coefficients that have no balancing or cobalancing numbers.
