Englishmen sitting in the cinema

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

Given nn seats,n∈I+∪n \in I^{+} \cup {00 }. And infinite Englishmen.

Prove that the number of ways of Englishmen sitting on the seats is Fn+2F_{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: FnF_{n} is the nth Fibonacci number, F1=1,F2=1,Fn=Fn−1+Fn−2F_{1} = 1, F_{2} = 1, F_{n} = F_{n-1} + F_{n-2} for n≥3n \geq 3.

#Combinatorics #FibonacciNumbers #Proofs

Note by Samuraiwarm Tsunayoshi
7 years, 4 months ago

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Comments

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 nn seats

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

Therefore, there're Fn+Fn+1=Fn+2F_{n} + F_{n+1} = F_{n+2} ways.

Samuraiwarm Tsunayoshi - 7 years, 4 months ago

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

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

Then,

Base Case: T0=F2=1,T1=F3=2 T_0 = F_2 = 1 , T_1 = F_3 = 2

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

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

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

⇒Tn+2=Fn+4 \Rightarrow T_{n+2} = F_{n+4}

Siddhartha Srivastava - 7 years, 4 months ago

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Thank you ^_^

Samuraiwarm Tsunayoshi - 7 years, 4 months ago
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