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# Englishmen sitting in the cinema

I just had a Thai national test yesterday, and I have a question about proving this.

Given $$n$$ seats,$$n \in I^{+} \cup$$ {$$0$$}. And infinite Englishmen.

Prove that the number of ways of Englishmen sitting on the seats is $$F_{n+2}$$, such that no two Englishmen sit in an adjacent seats (no one sitting is also counted as 1 way).

Example. (0 is vacant. 1 is occupied)

n=0; $$\varnothing \rightarrow$$ 1 way (no seats, no Englishmen)

n=1; 0 1$$\rightarrow$$ 2 ways

n=2; 00 01 10 $$\rightarrow$$ 3 ways

n=3; 000 100 010 001 101 $$\rightarrow$$ 5 ways

n=4; 0000 1000 0100 0010 0001 1010 1001 0101 $$\rightarrow$$ 8 ways

n=5; 00000 10000 01000 00100 00010 00001 10100 10010 10001 01010 01001 00101 10101 $$\rightarrow$$ 13 ways

n=6; 000000 100000 010000 001000 000100 000010 000001 101000 100100 100010 100001 010100 010010 010001 001010 001001 000101 101010 101001 100101 010101$$\rightarrow$$ 21 ways

etc.....

Note: $$F_{n}$$ is the nth Fibonacci number, $$F_{1} = 1, F_{2} = 1, F_{n} = F_{n-1} + F_{n-2}$$ for $$n \geq 3$$.

Note by Samuraiwarm Tsunayoshi
3 years, 1 month ago

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I tried proving by strong induction. I can do the induction step, but I can't start the basis step. T__T

I use double counting for the induction step.

Find the number of ways given $$n$$ seats

1st: nth position is vacant, (n-1) seats remaining, so there're $$F_{n+1}$$ ways of sitting.

2nd: nth position is occupied, so (n-1)th seat can't have anyone sitting, (n-2) seats remaining, so there're $$F_{n}$$ ways of sitting.

Therefore, there're $$F_{n} + F_{n+1} = F_{n+2}$$ ways. · 3 years, 1 month ago

Let number of ways of seating $$x$$ people be $$T_x$$

To Prove:- $$T_n = F_{n+2}$$

Then,

Base Case: $$T_0 = F_2 = 1 , T_1 = F_3 = 2$$

Inductive case : If true for $$n, n+1$$,

Then, $$T_{n} + T_{n+1} = T_{n+2}$$ ( Proved in Original Post)

Or, $$T_{n} + T_{n+1} = F_{n+2} + F_{n+3} = F_{n+4}$$

$$\Rightarrow T_{n+2} = F_{n+4}$$ · 3 years, 1 month ago