|
%I #40 Sep 08 2022 08:45:37
%S 0,1,6,61,660,8901,133266,2303865,43808040,928665225,21386693790,
%T 537861526965,14540730176700,423407835413325,13140639311294250,
%U 434929825450371825,15237733330856005200,565064979900590948625,22056613209702152061750,905913636742121921038125
%N Numerators of n-th approximants of a continued fraction for Pi-3.
%C The corresponding denominators are A001879(n), n >= 0.
%C 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.
%C For the approximants in lowest terms cf. the ones for 3*(Pi-3) given by A130411(n)/A130412(n) in lowest terms.
%C 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
%D J.-P. Delahaye, Le fascinant nombre pi, Pour la Science, Paris 1997. In German: Pi - die Story, Birkhäuser, 1999 Basel, p. 87.
%H B. C. Berndt, R. L. Lamphere, and B. M. Wilson <a href="http://projecteuclid.org/euclid.rmjm/1250127204">Chapter 12 of Ramanujan's second notebook: Continued fractions</a>, Rocky Mountain Journal of Mathematics, Volume 15, Number 2 (1985), 235-310
%H L. Euler, <a href="http://eulerarchive.maa.org/">De fractionibus continuis observationes</a>, The Euler Archive, Index Number 123, Section 67.
%H Wolfdieter Lang, <a href="/A142970/a142970_1.txt">Approximants for Pi-3 and more</a>
%H L. J. Lange, <a href="http://www.jstor.org/stable/2589152">An Elegant Continued Fraction for π</a>, The American Mathematical Monthly, 106 (1999), 456-458.
%F a(n) = 6*a(n-1) + ((2*n-1)^2)*a(n-2), a(0)=0, a(1)=1.
%F 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).
%F a(n) ~ (Pi-3) * 2^(n+3/2) * n^(n+2) / exp(n). - _Vaclav Kotesovec_, Oct 05 2013
%e Approximants a(n)/A001879(n) (not in lowest terms): [0/1]; [1/6]; [6/45]; [61/420]; [660/4725]; [8901/62370];..
%e Approximants in lowest terms: [0/1]; [1/6]; [2/15]; [61/420]; [44/315]; [989/6930]; ...
%t 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 *)
%o (Magma) 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
%Y Cf. A000796, A001879, A130411, A130412, A254795.
%K nonn,easy,frac,cofr
%O 0,3
%A _Wolfdieter Lang_, Sep 15 2008
|
