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%I M3097 N1255 #104 Feb 16 2025 08:32:25
%S 0,1,3,22,333,355,103993,104348,208341,312689,833719,1146408,4272943,
%T 5419351,80143857,165707065,245850922,411557987,1068966896,2549491779,
%U 6167950454,14885392687,21053343141,1783366216531,3587785776203,5371151992734,8958937768937
%N Numerators of convergents to Pi.
%C From _Alexander R. Povolotsky_, Apr 09 2012: (Start)
%C K. S. Lucas found, by brute-force search, using Maple programming, several different variants of integral identities which relate each of several first Pi convergents (A002485(n)/A002486(n)) to Pi.
%C I conjecture the following identity below, which represents a generalization of Stephen Lucas's experimentally obtained identities:
%C (-1)^n*(Pi-A002485(n)/A002486(n)) = (1/abs(i)*2^j)*Integral_{x=0..1} (x^l*(1-x)^m*(k+(k+i)*x^2)/(1+x^2)) dx where {i, j, k, l, m} are some integers (see the Mathematics Stack Exchange link below).
%C (End)
%C From a(1)=1 on also: Numbers for which |tan x| decreases monotonically to zero, in the same spirit as A004112, A046947, ... - _M. F. Hasler_, Apr 01 2013
%C See also A332095 for n*|tan n| < 1. - _M. F. Hasler_, Sep 13 2020
%D P. Beckmann, A History of Pi. Golem Press, Boulder, CO, 2nd ed., 1971, p. 171 (but beware errors).
%D CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 88.
%D P. Finsler, Über die Faktorenzerlegung natuerlicher Zahlen, Elemente der Mathematik, 2 (1947), 1-11, see p. 7.
%D K. H. Rosen et al., eds., Handbook of Discrete and Combinatorial Mathematics, CRC Press, 2000; p. 293.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Daniel Mondot, <a href="/A002485/b002485.txt">Table of n, a(n) for n = 0..1947</a> (terms 0..201 from T. D. Noe, terms 202..1000 from G. C. Greubel).
%H E. B. Burger, <a href="http://www.maa.org/sites/default/files/pdf/upload_library/22/Chauvenet/Burger.pdf">Diophantine Olympics and World Champions: Polynomials and Primes Down Under</a>, Amer. Math. Monthly, 107 (Nov. 2000), 822-829.
%H Marc Daumas, <a href="http://www.ipsl.jussieu.fr/~omamce/SP/Oct00/Marc_Daumas.pdf">Des implantations differentes ...</a>, see p. 8. [Broken link]
%H Henryk Fuks, <a href="http://arxiv.org/abs/1111.1739">Adam Adamandy Kochanski's approximations of Pi: reconstruction of the algorithm</a>, arXiv preprint arXiv:1111.1739 [math.HO], 2011-2014. Math. Intelligencer, Vol. 34 (No. 4), 2012, pp. 40-45.
%H S. K. Lucas,<a href="http://www.math.jmu.edu/~lucassk/Papers">Integral approximations to Pi with nonnegative integrands</a>
%H Mathematics Stack Exchange, <a href="http://math.stackexchange.com/questions/1956/">Is there an integral that proves pi > 333/106</a>
%H G. P. Michon, <a href="http://www.numericana.com/answer/fractions.htm">Continued Fractions</a>
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Pi.html">Pi</a>
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PiContinuedFraction.html">Pi Continued Fraction</a>
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PiApproximations.html">Pi Approximations</a>
%H <a href="/index/Ph#Pi314">Index entries for sequences related to the number Pi</a>
%e The convergents are 0, 1, 3, 22/7, 333/106, 355/113, 103993/33102, 104348/33215, 208341/66317, 312689/99532, 833719/265381, 1146408/364913, 4272943/1360120, 5419351/1725033, 80143857/25510582, 165707065/52746197, 245850922/78256779, 411557987/131002976, 1068966896/340262731, 2549491779/811528438, ... = A002485/A002486
%p Digits := 60: E := Pi; convert(evalf(E),confrac,50,'cvgts'): cvgts;
%t Join[{0, 1}, Numerator @ Convergents[Pi,29]] (* _Jean-François Alcover_, Apr 08 2011 *)
%o (PARI) contfracpnqn(cf=contfrac(Pi),#cf)[1,] \\ _M. F. Hasler_, Apr 01 2013, simplified Oct 13 2020
%o (PARI) e=9e9;for(n=1,1e9,abs(tan(n))<e && !print1(n",") && e=abs(tan(n))) \\ Illustration of |tan a(n)| -> 0 monotonically. - _M. F. Hasler_, Apr 01 2013
%Y Cf. A002486 (denominators), A046947, A072398/A072399.
%Y Cf. A096456 (numerators of convergents to Pi/2).
%K nonn,easy,nice,frac,changed
%O 0,3
%A _N. J. A. Sloane_
%E Extended and corrected by _David Sloan_, Sep 23 2002