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F_n sequences

Source: 1966-67 Germany R4 12.2 https://artofproblemsolving.com/community/c3208025_

October 13, 2024
combinatoricsalgebraSequence

Problem Statement

Let n≠0n \ne 0 be a natural number. A sequence of numbers is briefly called a sequence “FnF_n” if nn different numbers z1z_1, z2z_2, ......, znz_n exist so that the following conditions are fulfilled: (1) Each term of the sequence is one of the numbers z1z_1, z2z_2, ......, znz_n. (2) Each of the numbers z1z_1, z2z_2, ......, znz_n occurs at least once in the sequence. (3) Any two immediately consecutive members of the sequence are different numbers. (4) No subsequence of the sequence has the form {a,b,a,b}\{a, b, a, b\} with a≠ba \ne b.
Note: A subsequence of a given sequence {x1,x2,x3,...}\{x_1, x_2, x_3, ...\} or {x1,x2,x3,...,xs}\{x_1, x_2, x_3, ..., x_s\} is called any sequence of the form {xm1,xm2,xm3,...}\{x_{m1}, x_{m2}, x_{m3}, ...\} or {xm1,xm2,xm3,...,xmt}\{x_{m1}, x_{m2}, x_{m3}, ..., x_{mt}\} with natural numbers m1<m2<m3<...m_1 < m_2 < m_3 < ...
Answer the following questions: a) Given nn, are there sequences FnF_n of arbitrarily long length? b) If question (a) is answered in the negative for an nn: What is the largest possible number of terms that a sequence FnF_n can have (given nn)?
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