MathDB

Let

R_1,R_2, \ldots

be the family of finite sequences of positive integers defined by the following rules: R_1 \equal{} (1), and if R_{n - 1} \equal{} (x_1, \ldots, x_s), then
R_n \equal{} (1, 2, \ldots, x_1, 1, 2, \ldots, x_2, \ldots, 1, 2, \ldots, x_s, n).
For example, R_2 \equal{} (1, 2), R_3 \equal{} (1, 1, 2, 3), R_4 \equal{} (1, 1, 1, 2, 1, 2, 3, 4). Prove that if

n > 1,

then the

k

th term from the left in

R_n

is equal to 1 if and only if the

k

th term from the right in

R_n

is different from 1.

Problem Statement