18 Apr '05 00:03

Consider a game played with n identical coins. You are to arrange the coins in either one row or two rows under the following rules:

1. The bottom row cannot have any gaps, i.e., any coin on the bottom row must touch its neighbor(s) if it has a neighbor or neighbors.

2. Any coin placed on the top row must touch two coins from the bottom row.

Let A(n) (for n > 0) be the number of distinguishable arrangements of these n coins allowed under the rules of the game.

Find a general expression for A(n). What sequence do the numbers {A(j)} produce for j = 0,1,2,3,…(for this, take A(0) = 0)?

1. The bottom row cannot have any gaps, i.e., any coin on the bottom row must touch its neighbor(s) if it has a neighbor or neighbors.

2. Any coin placed on the top row must touch two coins from the bottom row.

Let A(n) (for n > 0) be the number of distinguishable arrangements of these n coins allowed under the rules of the game.

Find a general expression for A(n). What sequence do the numbers {A(j)} produce for j = 0,1,2,3,…(for this, take A(0) = 0)?