A130412 -id:A130412 - 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

A130411 Numerator of partial sums of a series for 3*(Pi-3).

Original entry on oeis.org

1, 2, 61, 44, 989, 6346, 51197, 36056, 4127401, 2057402, 189721879, 236723324, 1422382919, 20600649518, 10227626700773, 638723926928, 1278290544991, 23635180313246, 94585786464329, 969106771716436, 83372817133541471

Offset: 1

Wolfdieter Lang, Jun 01 2007, Sep 09 2008, Oct 06 2008

Denominators are given in A130412.
The rationals (in lowest terms) r(n):=3*sum(((-1)^(j+1))/(j*(j+1)*(2*j+1)),j=1..n) have the limit 3*(Pi-3), approximately 0.424777962, for n->infinity.
These partial sums result from those for the more familiar series s(n):=sum(((-1)^(j+1))/(2*j*(2*j+1)*(2*j+2)),j=1..n) with limit (Pi-3)/4 which is approximately 0.0353981635. r(n)= 12*s(n). This series is attributed to K. G. Nilakantha, see, e.g., the R. Roy reference. eq.(13).
The sum r(n)/3 gives the n-th approximant to the continued fraction 1^2/(6+3^2/(6+5^2/6+...Proof with Euler's 1748 conversion of continued fractions into series. The denominators q(n)=A001879 of the n-th approximant of this continued fraction is used. The author (WL) reconsidered this entry after an e-mail from R. Rosenthal Jul 16 2008 pointing out the Pi-3 continued fraction.

			Rationals r(n), n>=1: [1/2, 2/5, 61/140, 44/105, 989/2310, 6346/15015, 51197/120120, ...].
Rationals s(n)=r(n)/12, n>=1: [1/24, 1/30, 61/1680, 11/315, 989/27720, 3173/90090, 51197/1441440, ...].

W. Lang, Rationals and limit.
Ranjan Roy, The Discovery of the Series Formula for Pi by Leibniz, Gregory and Nilakantha, Math. Magazine 63 (1990), 291-306. Reprinted in: Pi: A Source Book, eds. L. Berggren, et al., Springer, New York, 1997, pp. 92-107.

a(n) = numerator(r(n)) with the rationals r(n) given above.

A130413 Numerators of partial sums for a series for Pi/3.

Original entry on oeis.org

1, 19, 47, 1321, 989, 21779, 141481, 1132277, 801821, 91424611, 45706007, 4205393539, 5256312899, 31539920369, 457304942543, 226832956041173, 14176557010703, 28353956712541, 524535004412921, 2098185082863029

Offset: 0

Wolfdieter Lang, Jun 01 2007

The denominators are given in A130414.
The rationals r(n) = 1 + (4/3)*Sum_{j=1..n} (-1)^(j+1)/((2*j+1)*((2*j+1)^2-1)), n >= 0, have the limit lim_{n->infinity} r(n) = Pi/3, approximately 1.047197551.
This series is obtained from the one for Pi/4 (attributed to Nilakantha) obtained by multiplication with 3/4. See the R. Roy link eq.(13).

			Rationals r(n): 1, 19/18, 47/45, 1321/1260, 989/945, 21779/20790, 141481/135135, ...

Robert Israel, Table of n, a(n) for n = 0..1000
W. Lang, Rationals and limit
Ranjan Roy, The Discovery of the Series Formula for Pi by Leibniz, Gregory and Nilakantha, Math. Magazine 63 (1990), 291-306.

Cf. A130411/A130412 (partial sums for a series of 3*(Pi-3)).

f:= n -> numer(1+ (4/3)*add(((-1)^(j+1))/((2*j+1)*((2*j+1)^2-1)),j=1..n)):
map(f, [$0..20]); # Robert Israel, Jul 27 2015

a(n) = numerator(r(n)), n >= 0, with r(n) defined above.

G.f. for r(n): 4*arctan(sqrt(x))/(3*sqrt(x)*(1-x)) - log(x+1)/(3*x). - Robert Israel, Jul 27 2015

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

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A130411 Numerator of partial sums of a series for 3*(Pi-3).

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A130413 Numerators of partial sums for a series for Pi/3.

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