%I #46 Jul 05 2024 08:12:00

%S 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,

%T 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,

%U 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

%N Decimal expansion of L^2/Pi where L is the lemniscate constant A062539.

%C 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.

%D O. Perron, Die Lehre von den Kettenbrüchen, Band II, Teubner, Stuttgart, 1957

%H Muniru A Asiru, <a href="/A254794/b254794.txt">Table of n, a(n) for n = 1..2000</a>

%H Peter Bala, <a href="/A096427/a096427.pdf">Notes on the constants A096427 and A224268 </a>

%H B. C. Berndt, R. L. Lamphere and B. M. Wilson, <a href="https://doi.org/10.1216/RMJ-1985-15-2-235">Chapter 12 of Ramanujan's second notebook: Continued fractions</a>, Rocky Mountain Journal of Mathematics, Volume 15, Number 2 (1985), 235-310.

%H T. J. Osler, <a href="https://www.jstor.org/stable/23248554">The missing fractions in Brouncker's sequence of continued fractions for Pi</a>, The Mathematical Gazette, 96(2012), pp. 221-225.

%F L^2/Pi = 2*( (1/4)!/(1/2)! )^4 = 9/4*( (1/4)!/(3/4)! )^2.

%F L^2/Pi = lim_{n -> oo} (4*n + 2) * Product {k = 0..n} ( (4*k - 1)/(4*k + 1) )^2

%F 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.

%F 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.

%F 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).

%F (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.

%F From _Peter Bala_, Feb 28 2019: (Start)

%F C = 2*A224268/A096427.

%F 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 ).

%F 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

%F 1/C = Product_{k = 0..m} (1 + 4*k)/(1 - 4*k) * Product_{n >= 0} ( 1 - (4*m + 2)^2/(4*n + 3)^2 ).

%F C = (Pi/2) * ( Sum_{n = -oo..oo} exp(-Pi*n^2) )^4. (End)

%F Equals A133748/Pi. - _Hugo Pfoertner_, Apr 13 2024

%e 2.18843961522647663883676994070446454325937272282556672211928621....

%p #A254794

%p digits:=105:

%p 2*( GAMMA(5/4)/GAMMA(3/2) )^4:

%p evalf(%);

%t RealDigits[2*(Gamma[5/4]/Gamma[3/2])^4, 10, 110][[1]] (* _G. C. Greubel_, Mar 06 2019 *)

%o (PARI) default(realprecision, 110); 2*(gamma(5/4)/gamma(3/2))^4 \\ _G. C. Greubel_, Mar 06 2019

%o (Magma) SetDefaultRealField(RealField(110)); 2*(Gamma(5/4)/Gamma(3/2))^4; // _G. C. Greubel_, Mar 06 2019

%o (Sage) numerical_approx(2*(gamma(5/4)/gamma(3/2))^4, digits=110) # _G. C. Greubel_, Mar 06 2019

%Y Cf. A000796, A062539, A254795, A254796, A096427, A133748, A224268.

%K cons,nonn,easy

%O 1,1

%A _Peter Bala_, Feb 22 2015