binPis' =  let 
    next ( x@( n, p, q , p', q' ) ) = let
        p'' = ( 2*n-1 ) * p' + ( n-1 ) ^2 * p
        q'' =  ( 2*n-1 ) * q'  + ( n-1 ) ^2 * q
        ( f, f' ) = ( div p q, div p' q' )
        g = foldl1 gcd [ p', q', p'', q'' ]
        [ np', nq', np'', nq'' ] = map ( flip div g ) [ p',q',p'',q'' ]
        in if f == f'
            then ( f,x ) : next ( n + 1, 2*(np'- f * nq') , nq', 2*(np'' - f * nq''), nq'' )
            else next ( n + 1, p',q', p'', q'' )
    in map fst $ next ( 2, 2, 3, 5, 3 )
 
binPis'' = zipWith (+) 
    ( zipWith ( * ) binPis' $ iterate (*(-1)) 1 )
    ( concat $ repeat [ 0, 1 ] )
 
main = do
    print $  take 100 binPis''
    print $ sum $ take 100 $ zipWith ( * )  [ 4*(-0.5)^n | n<-[0..] ] $ map ( fromInteger ) $ binPis''
 

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