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$\pi=\varpi_2$ can be considered as a member of the sequence of real numbers $$\varpi_m=2\int\limits_0^1\frac{dt}{\sqrt{1-t^m}},$$ and, for example, the Wallis product formula $$\pi=4\prod_{n=1}^\infty{\frac{2n(2n+2)}{(2n+1)^2}}$$ can be generalized to all these m-clover constants (http://arxiv.org/abs/1212.4178 A Wallis Product on Clovers, by Trevor Hyde): $$\varpi_m = \frac{2(m+2)}{m}\prod_{n=1}^\infty{\frac{2n(2mn+ m +2)} {(2n+1)(2mn+2)}}.$$ On the other hand, for $\pi$ there exists the well known Dalzel's nice integral formula $$\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 \frac{dx}{1+x^2},$$ (see the discussion: Source and context of $\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 dx/(1+x^2)$? ).

I'm curious whether something like the Dalzel's formula exists, for example, for the lemniscate constant $\varpi_4$.

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One thing that makes the Dalzel formula nice is that $22/7$ is such a good approximation to $\pi$. Do you know a good rational approximation to the lemniscate constant? –  Gerry Myerson Aug 26 '13 at 12:59
    
Here is the first hundred terms of the continued fraction for $\omega_2/2$ (so just the integral, without the factor of two): {1, 3, 4, 1, 1, 1, 5, 2, 1, 4, 1, 6, 1, 1, 4, 4, 3, 4, 4, 1, 1, 1, 1, \ 21, 1, 9, 1, 3, 1, 2, 11, 1, 1, 1, 5, 6, 8, 1, 2, 1, 168, 1, 2, 1, 1, \ 3, 1, 2, 1, 1, 1, 2, 1, 3, 6, 2, 1, 1, 19, 3, 1, 43, 5, 2, 1, 1, 1, \ 3, 1, 1, 3, 1, 4, 1, 4, 1, 19, 1, 5, 3, 1, 3, 1, 4, 1, 3, 2, 1, 40, \ 2, 1, 5, 9, 4, 6, 2, 1, 3, 1, 1} –  Igor Rivin Aug 26 '13 at 14:33
    
The backslashes are linebreaks, sorry. In any case, it seems that to get an unusually good approximation, you have to go to the 168 term. –  Igor Rivin Aug 26 '13 at 14:34
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You might be interested in the following paper nieuwarchief.nl/serie5/pdf/naw5-2000-01-4-372.pdf which describes (see page 7), more general Dalzel like formulae for $\pi$ (which essentially include the Dalzel formula as the "simplest" case) –  Suvrit Aug 26 '13 at 16:38
    
For example, 21/8 approximates $\varpi_4$ with relative accuracy $1.1\cdot 10^{-3}$, and 97/37 -- with relative accuracy $1.7\cdot 10^{-4}$. 22/7 approximates $\pi$ with relative accuracy $4.0\cdot 10^{-4}$. Beukers' article is indeed interesting and informative. Interestingly, it seems 17/7 approximates $\varpi_6\approx 2.42865$ with relative accuracy $3.3\cdot 10^{-5}$! –  Zurab Silagadze Aug 27 '13 at 6:13
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