A254794 Decimal expansion of L^2/Pi where L is the lemniscate constant A062539.

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

Offset

1,1

Comments

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.

References

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

Links

Muniru A Asiru, Table of n, a(n) for n = 1..2000

Peter Bala, Notes on the constants A096427 and A224268

B. C. Berndt, R. L. Lamphere and B. M. Wilson, Chapter 12 of Ramanujan's second notebook: Continued fractions, Rocky Mountain Journal of Mathematics, Volume 15, Number 2 (1985), 235-310.

T. J. Osler, The missing fractions in Brouncker's sequence of continued fractions for Pi, The Mathematical Gazette, 96(2012), pp. 221-225.

Formula

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

L^2/Pi = lim_{n -> oo} (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.

From Peter Bala, Feb 28 2019: (Start)

C = 2*A224268/A096427.

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 = -oo..oo} exp(-Pi*n^2) )^4. (End)

Equals A133748/Pi. - Hugo Pfoertner, Apr 13 2024

Example

2.18843961522647663883676994070446454325937272282556672211928621....

Maple

#A254794
digits:=105:
2*( GAMMA(5/4)/GAMMA(3/2) )^4:
evalf(%);

Mathematica

RealDigits[2*(Gamma[5/4]/Gamma[3/2])^4, 10, 110][[1]] (* G. C. Greubel, Mar 06 2019 *)

Prog

(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

Crossrefs

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

Sequence in context: A123516 A193604 A016446 * A086657 A188922 A036296

Adjacent sequences: A254791 A254792 A254793 * A254795 A254796 A254797

Keyword

cons,nonn,easy

Author

Peter Bala, Feb 22 2015

Status

approved