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Benoit Cloitre

## and in a mirror

Benoit Cloitre, still very creative, is pursuing his chronicles about resemblance between famous constants, after those between   et .

### 1 For

Let and then . Thus, if we write down

then we obtain an inverse Brounker-like continued fraction.

### 2 For

Let and then . The equivalent continued fraction is

### Proof for Pi

It is easy to see by induction that and we recognise the Wallis product.

### Proof for

Let . We easily see that for , .

### Checking under the software Pari-GP

For Pi

x=171679;u=x;for(n=2,100,u=n*(n-1)/u+1;if(n%2==0,print1((u-n-1)/(u-n)-prod(k=1,n/2,4*k^2/(4*k^2-1))*(1+1/n)*(x-1)/(x-2),",")))

,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,

For log(2)

f(n)=if(n<2,171679,(n-1)^2/f(n-1)+1);Q(n)=1/(2*n-f(2*n))-1/(171679-1);for(n=1,10,print1(Q(n+1)-sum(k=1,2*n+1,(-1)^(k+1)/k),","))

0,0,0,0,0,0,0,0,0,0,