MathDB

One country consists of islands

A_1,A_2,\cdots,A_N

,The ministry of transport decided to build some bridges such that anyone will can travel by car from any of the islands

A_1,A_2,\cdots,A_N

to any another island by one or more of these bridges. For technical reasons the only bridges that can be built is between

A_i

and

A_{i+1}

where

i = 1,2,\cdots,N-1

, and between

A_i

and

A_N

where

i<N

.
We say that a plan to build some bridges is good if it is satisfies the above conditions , but when we remove any bridge it will not satisfy this conditions. We assume that there is

a_N

of good plans. Observe that

a_1 = 1

(The only good plan is to not build any bridge) , and

a_2 = 1

(We build one bridge).
1-Prove that

a_3 = 3

2-Draw at least

5

different good plans in the case that

N=4

and the islands are the vertices of a square 3-Compute

a_4

4-Compute

a_6

5-Prove that there is a positive integer

i

such that

1438

divides

a_i

2

Problems(1)