A254795 -id:A254795 - cp's OEIS frontend

A142970 Numerators of n-th approximants of a continued fraction for Pi-3.

0, 1, 6, 61, 660, 8901, 133266, 2303865, 43808040, 928665225, 21386693790, 537861526965, 14540730176700, 423407835413325, 13140639311294250, 434929825450371825, 15237733330856005200, 565064979900590948625, 22056613209702152061750, 905913636742121921038125

Offset: 0

Wolfdieter Lang, Sep 15 2008

The corresponding denominators are A001879(n), n >= 0.
Pi = 3 + 1^2/(6 + 3^2/(6 + 5^2/(6 + ... ))). See the J.-P. Delahaye reference. R. Rosenthal mentioned this continued fraction in an e-mail to the author Jul 16 2008.
For the approximants in lowest terms cf. the ones for 3*(Pi-3) given by A130411(n)/A130412(n) in lowest terms.
The above continued fraction for Pi is the particular case n = 0, x = 3 of a result of Ramanujan, previously given by Euler - see Berndt et al., Chapter 12, Entry 25, p. 268. - Peter Bala, Feb 19 2015

			Approximants a(n)/A001879(n) (not in lowest terms): [0/1]; [1/6]; [6/45]; [61/420]; [660/4725]; [8901/62370];..
Approximants in lowest terms: [0/1]; [1/6]; [2/15]; [61/420]; [44/315]; [989/6930]; ...

J.-P. Delahaye, Le fascinant nombre pi, Pour la Science, Paris 1997. In German: Pi - die Story, Birkhäuser, 1999 Basel, p. 87.

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
L. Euler, De fractionibus continuis observationes, The Euler Archive, Index Number 123, Section 67.
Wolfdieter Lang, Approximants for Pi-3 and more
L. J. Lange, An Elegant Continued Fraction for π, The American Mathematical Monthly, 106 (1999), 456-458.

Cf. A000796, A001879, A130411, A130412, A254795.

I:=[1,6]; [0] cat [n le 2 select I[n] else 6*Self(n-1)+(2*n-1)^2*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 20 2015

RecurrenceTable[{a[0]==0, a[1]==1, a[n]==6 a[n-1] + (2 n-1)^2 a[n-2]}, a, {n, 30}] (* Vincenzo Librandi, Feb 20 2015 *)

a(n) = 6*a(n-1) + ((2*n-1)^2)*a(n-2), a(0)=0, a(1)=1.
E.g.f.: (-3*(1+x-sqrt(1-4*x^2))+ 2*(1+x)*arcsin(2*x))/(1-2*x)^(5/2) from the solution of the linear second order differential equation (1-4*x^2)*y''(x) - 2*(8*x+3)*y'(x) - 9*y(x)=0, obtained from the recurrence, with inputs y(0)=0 and y'(0)=1. A special solution is the e.g.f. of the denominators A001879: (1+x)/(1-2*x)^(5/2).
a(n) ~ (Pi-3) * 2^(n+3/2) * n^(n+2) / exp(n). - Vaclav Kotesovec, Oct 05 2013

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

Original entry on oeis.org

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

A254796 Denominators of the convergents of the generalized continued fraction 2 + 1^2/(4 + 3^2/(4 + 5^2/(4 + ... ))).

Original entry on oeis.org

1, 4, 25, 200, 2025, 24300, 342225, 5475600, 98903025, 1978060500, 43616234025, 1046789616600, 27260146265625, 763284095437500, 22925783009390625, 733625056300500000, 24966177697226390625, 898782397100150062500, 34178697267502928765625

Offset: 0

Peter Bala, Feb 23 2015

The generalized continued fraction 2 + 1^2/(4 + 3^2/(4 + 5^2/(4 + ... ))) represents the constant L^2/Pi = 2.188439... = A254794, where L is the lemniscate constant A062539. See A254795 for the numerators of the convergents of the continued fraction.

			54/25 = 2.16, 441/200 = 2.205 etc approach 2.188..

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

Cf. A007696, A024199, A062539, A254794, A254795.

I:=[1,4]; [n le 2 select I[n] else 4*Self(n-1)+(2*n-3)^2*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Feb 24 2015

a[0] := 1: a[1] := 4:
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);

RecurrenceTable[{a[0] == 1, a[1] == 4, a[n] == 4 a[n - 1] + (2 n - 1)^2 a[n - 2]}, a, {n, 20}] (* Vincenzo Librandi, Feb 24 2015 *)

a(2*n) = A007696(n+1)^2 = ( Product {k = 0..n} 4*k + 1 )^2.
a(2*n-1) = 4*n*A007696(n)^2 = 4*n * ( Product {k = 0..n-1} 4*k + 1 )^2.
a(n) = 4*a(n-1) + (2*n - 1)^2*a(n-2) with a(0) = 1, a(1) = 4.
a(2*n+1) = 4*(n + 1)*a(2*n); a(2*n) = (4*n + 2)*a(2*n-1) + a(2*n-2).
Empirical e.g.f.: ((-Q(1/2, -3)-Q(-1/2, -3))*P(1/2, (2*x+3)/(2*x-1))+Q(1/2, (2*x+3)/(2*x-1))*(P(1/2, -3)+P(-1/2, -3)))/((1-2*x)^(3/2)*(-Q(-1/2, -3)*P(1/2, -3)+Q(1/2, -3)*P(-1/2, -3))) where P and Q are Legendre functions of the first and second kinds. - Robert Israel, Feb 24 2015

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A142970 Numerators of n-th approximants of a continued fraction for Pi-3.

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

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A254796 Denominators of the convergents of the generalized continued fraction 2 + 1^2/(4 + 3^2/(4 + 5^2/(4 + ... ))).

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