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A254794
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Decimal expansion of L^2/Pi where L is the lemniscate constant A062539.
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3
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2, 1, 8, 8, 4, 3, 9, 6, 1, 5, 2, 2, 6, 4, 7, 6, 6, 3, 8, 8, 3, 6, 7, 6, 9, 9, 4, 0, 7, 0, 4, 4, 6, 4, 5, 4, 3, 2, 5, 9, 3, 7, 2, 7, 2, 2, 8, 2, 5, 5, 6, 6, 7, 2, 2, 1, 1, 9, 2, 8, 6, 2, 1, 0, 5, 7, 9, 4, 5, 1, 9, 3, 8, 4, 4, 5, 9, 3, 2, 9, 4, 7, 7, 7, 1, 0, 3, 3, 1, 4, 9, 6, 7, 7, 5, 6, 0, 8, 6, 3, 1, 8, 0, 2
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OFFSET
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1,1
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COMMENTS
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Brouncker gave the generalized continued fraction expansion 4/Pi = 1 + 1^2/(2 + 3^2/(2 + 5^2/(2 + ... ))). More generally, Osler shows that the continued fraction n + 1^2/(2*n + 3^2/(2*n + 5^2/(2*n + ... ))) equals a rational multiple of 4/Pi or its reciprocal when n is a positive odd integer, and equals a rational multiple of L^2/Pi or its reciprocal when n is a positive even integer.
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REFERENCES
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O. Perron, Die Lehre von den Kettenbrüchen, Band II, Teubner, Stuttgart, 1957
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LINKS
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FORMULA
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L^2/Pi = 2*( (1/4)!/(1/2)! )^4 = 9/4*( (1/4)!/(3/4)! )^2.
L^2/Pi = limit (n -> inf) (4*n + 2) * Product {k = 0..n} ( (4*k - 1)/(4*k + 1) )^2
Generalized continued fraction: L^2/Pi = 2 + 1^2/(4 + 3^2/(4 + 5^2/(4 + ... ))). This is the particular case n = 0, x = 2 of a result of Ramanujan - see Berndt et al., Entry 25. See also Perron, p. 35.
The sequence of convergents to Ramanujan's continued fraction begins [2/1, 9/4, 54/25, 441/200, 4410/2025, ...]. See A254795 for the numerators and A254796 for the denominators.
Another continued fraction is L^2/Pi = 1 + 2/(1 + 1*3/(2 + 3*5/(2 + 5*7/(2 + 7*9/(2 + ... ))))), which can be transformed into the slowly converging series: L^2/Pi = 1 + 4 * Sum {n >= 0} P(n)^2/(4*n + 5), where P(n) = Product {k = 1..n} (4*k - 1)/(4*k + 1).
(L^2/Pi)^2 = 3 + 2*( 1^2/(1 + 1^2/(3 + 3^2/(1 + 3^2/(3 + 5^2/(1 + 5^2/(3 + ... )))))) ) follows by setting n = 0, x = 2 in Entry 26 of Berndt et al.
For m = 0,1,2,..., C = 4*(m + 1)*P(m)/Q(m), where P(m) = Product_{n >= 1} ( 1 - (4*m + 3)^2/(4*n + 1)^2 ) and Q(m) = Product_{n >= 0} ( 1 - (4*m + 1)^2/(4*n + 3)^2 ).
For m = 0,1,2,..., C = - Product_{k = 1..m} (1 - 4*k)/(1 + 4*k) * Product_{n >= 0} ( 1 - (4*m + 2)^2/(4*n + 1)^2 ) and
1/C = Product_{k = 0..m} (1 + 4*k)/(1 - 4*k) * Product_{n >= 0} ( 1 - (4*m + 2)^2/(4*n + 3)^2 ).
C = (Pi/2) * ( Sum_{n = -inf..inf} exp(-Pi*n^2) )^4. (End)
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EXAMPLE
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2.18843961522647663883676994070446454325937272282556672211928621....
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MAPLE
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digits:=105:
2*( GAMMA(5/4)/GAMMA(3/2) )^4:
evalf(%);
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MATHEMATICA
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RealDigits[2*(Gamma[5/4]/Gamma[3/2])^4, 10, 110][[1]] (* G. C. Greubel, Mar 06 2019 *)
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PROG
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(PARI) default(realprecision, 110); 2*(gamma(5/4)/gamma(3/2))^4 \\ G. C. Greubel, Mar 06 2019
(Magma) SetDefaultRealField(RealField(110)); 2*(Gamma(5/4)/Gamma(3/2))^4; // G. C. Greubel, Mar 06 2019
(Sage) numerical_approx(2*(gamma(5/4)/gamma(3/2))^4, digits=110) # G. C. Greubel, Mar 06 2019
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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