摘要
The purpose of this paper is to consider the (Formula presented.)-center problem with collinear centers, to identify its syzygy sequences that can be realized by minimizers of the Lagrangian action functional and to count the number of such syzygy sequences. In particular, we show that the number of such realizable syzygy sequences of length (Formula presented.) greater than or equal to two for the 3-center problem is at least (Formula presented.), where (Formula presented.) is the Fibonacci sequence. Moreover, with fixed length (Formula presented.), the density of such realizable syzygy sequences of length (Formula presented.) for the (Formula presented.)-center problem approaches one as (Formula presented.) increases to infinity. Using reflection symmetry, the minimizers that we found can be extended to periodic solutions.