A254796 -id:A254796 - cp's OEIS frontend

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

Peter Bala, Feb 22 2015

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.

			2.18843961522647663883676994070446454325937272282556672211928621....

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

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.

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

SetDefaultRealField(RealField(110)); 2*(Gamma(5/4)/Gamma(3/2))^4; // G. C. Greubel, Mar 06 2019

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

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

default(realprecision, 110); 2*(gamma(5/4)/gamma(3/2))^4 \\ G. C. Greubel, Mar 06 2019

numerical_approx(2*(gamma(5/4)/gamma(3/2))^4, digits=110) # G. C. Greubel, Mar 06 2019

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

A254795 Numerators of the convergents of the generalized continued fraction 2 + 1^2/(4 + 3^2/(4 + 5^2/(4 + ... ))).

Original entry on oeis.org

2, 9, 54, 441, 4410, 53361, 747054, 12006225, 216112050, 4334247225, 95353438950, 2292816782025, 59613236332650, 1671463434096225, 50143903022886750, 1606276360166472225, 54613396245660055650, 1967688541203928475625, 74772164565749282073750

Offset: 0

Peter Bala, Feb 23 2015

Brouncker gave the generalized continued fraction expansion 4/Pi = 1 + 1^2/(2 + 3^2/(2 + 5^2/(2 + ... ))). The sequence of convergents begins [1/1, 3/2, 15/13, 105/76, ... ]. The numerators of the convergents are in A001147, the denominators in A024199.

In extending Brouckner's result, Osler showed that 2 + 1^2/(4 + 3^2/(4 + 5^2/(4 + ... ))) = L^2/Pi, where L is the lemniscate constant A062539. The sequence of convergents to Osler's continued fraction begins [2/1, 9/4, 54/25, 441/200, 4410/2025, ...]. Here we list the (unreduced) numerators of these convergents. See A254796 for the sequence of denominators. See A254794 for the decimal expansion of L^2/Pi.

T. J. Osler, The missing fractions in Brouncker’s sequence of continued fractions for Pi

Cf. A008545, A001147, A024199, A142970, A254794, A254796.

a[0] := 2: a[1] := 9:
for n from 2 to 18 do a[n] := 4*a[n-1] + (2*n-1)^2*a[n-2] end do:
seq(a[n], n = 0 .. 18);

a(2*n-1) = ( A008545(n) )^2 = ( Product {k = 0..n-1} 4*k + 3 )^2.
a(2*n) = (4*n + 2)*( A008545(n) )^2 = (4*n + 2)*( Product {k = 0..n-1} 4*k + 3 )^2.
a(n) = 4*a(n-1) + (2*n - 1)^2*a(n-2) with a(0) = 2, a(1) = 9.
a(2*n) = (4*n + 2)*a(2*n-1); a(2*n+1) = (4*n + 4)*a(2*n) + a(2*n-1).

cp's OEIS Frontend

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

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