(k,q)-平坦数列研究

A Type of Special Circular Sequence:(k,q)-flat Sequence

一个环形数列是指将前n个正整数排成一圈。对于一个给定的环形数列,用S(n,k,q)表示其所有连续k项和的q次方的和,LS(n,k,q)记为所有S(n,k,q)中的最小值。当一个环形数列的S(n,k,q)值达到LS(n,k,q)时,称此数列为(k,q)-平坦数列。给出了(k,q)-平坦数列几个性质,并给出了(2,2)-平坦数列的证明和LS(n,2,2)的值。借助于计算机,找出了7≤n≤12,3≤k≤5的(k,q)-平坦数列。

A circular sequence is a circular arrangement of the first n positive integers. Given any circular sequence, let S(n,k,q) be the sum of qth powers of every sum of k consecutive terms. LS(n,k,q) is defined as the minimum sum of all possible S(n,k,q). A (k,q)-flat sequence is a circular sequence where S(n,k,q) is LS(n,k,q). This paper gives some properties of (k,q)-flat sequence and LS(n,2,2) with (2,2)-flat sequence. By means of computer, it gives some (k,2)-flat sequence for 7≤n≤12,3≤k≤5.