[r-t] Wraps of Rounds
Alexander Holroyd
holroyd at math.ubc.ca
Wed Nov 10 23:14:52 UTC 2004
It simplifies a bit more, I think.
Let f(n) be the number of possible place notations on n bells (including
rounds). Thus
f(0)=1
f(1)=1
f(2)=2
f(3)=3
f(4)=5 ....
They are indeed Fibonacci numbers (but you have to be careful about how
you number them!)
We have f(0)+f(1)+...+f(n) = f(n+2)-1 (can prove by induction).
The number of possible forward wraps on 2n bells (including rounds) is
W(2n) = f(0)+f(1)+...+f(n-1)+f(n)+f(n-1)+...+f(1)
= [f(0)+...+f(n)]+[f(0)+...+f(n-1)]-f(0)
= f(n+2)-1+f(n+1)-1-1
= f(n+3)-3
Thus
W(2) = 2
W(4) = 5
W(6) = 10
W(8) = 18
W(10) = 31
W(12) = 52
Ander
On Wed, 10 Nov 2004, Mark Davies wrote:
> I wrote,
>
> > On 2N bells, the number of wraps is 1+f(N)+2[f(1)+...+f(N-1)], where
> > f(x) = the number of possible place notations on x bells = fibonacci.
>
> Should have said, since sums of Fibonacci series are Fibonacci numbers less
> one, this simplifies to:
>
> Wraps on 2N bells = f(N) + 2f(N+1) - 3.
>
> Use the golden ratio to find a closed formula for f(N) if you want. Note
> f(N) = fib(N-1).
>
> Changeringing is not just about group theory, hoorah. :-D
>
> MBD
>
>
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