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 A002485 Numerators of convergents to Pi. (Formerly M3097 N1255) 33
 0, 1, 3, 22, 333, 355, 103993, 104348, 208341, 312689, 833719, 1146408, 4272943, 5419351, 80143857, 165707065, 245850922, 411557987, 1068966896, 2549491779, 6167950454, 14885392687, 21053343141, 1783366216531, 3587785776203, 5371151992734, 8958937768937 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS From Alexander R. Povolotsky, Apr 09 2012: (Start) 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. I conjecture the following identity below, which represents a generalization of Stephen Lucas' experimentally obtained identities:   (-1)^n*(Pi-A002485(n)/A002486(n)) = 1/abs(i)*2^j)*Integrate(x^l*(1-x)^m*(k+(k+i)*x^2)/(1+x^2),x= 0..1) where {i, j, k, l, m} are some integers (see the StackExchange link below). (End) 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 REFERENCES P. Beckmann, A History of Pi. Golem Press, Boulder, CO, 2nd ed., 1971, p. 171 (but beware errors). CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 88. P. Finsler, Über die Faktorenzerlegung natuerlicher Zahlen, Elemente der Mathematik, 2 (1947), 1-11, see p. 7. K. H. Rosen et al., eds., Handbook of Discrete and Combinatorial Mathematics, CRC Press, 2000; p. 293. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..201 from T. D. Noe) E. B. Burger, Diophantine Olympics and World Champions: Polynomials and Primes Down Under, Amer. Math. Monthly, 107 (Nov. 2000), 822-829. Marc Daumas, Des implantations differentes ..., see p. 8. Henryk Fuks, Adam Adamandy Kochanski's approximations of pi: reconstruction of the algorithm, arXiv preprint arXiv:1111.1739, 2011. Math. Intelligencer, Vol. 34 (No. 4), 2012, pp. 40-45. G. P. Michon, Continued Fractions StackExchange, Is there an integral that proves pi > 333/106 Eric Weisstein's World of Mathematics, Pi. Eric Weisstein's World of Mathematics, Pi continued fraction. Eric Weisstein's World of Mathematics, Pi Approximations EXAMPLE 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 MAPLE Digits := 60: E := Pi; convert(evalf(E), confrac, 50, 'cvgts'): cvgts; MATHEMATICA Join[{0, 1}, Numerator @ Convergents[Pi, 29]] (* Jean-François Alcover, Apr 08 2011 *) PROG (PARI) for(i=1, #cf=contfrac(Pi), print1(contfracpnqn(vecextract(cf, 2^i-1))[1, 1]", ")) \\ - M. F. Hasler, Apr 01 2013 (PARI) e=9e9; for(n=1, 1e9, abs(tan(n)) 0 monotonically. - M. F. Hasler, Apr 01 2013 CROSSREFS Cf. A002486 (denominators), A046947, A072398/A072399. Sequence in context: A189897 A306578 A046947 * A193193 A099750 A219268 Adjacent sequences:  A002482 A002483 A002484 * A002486 A002487 A002488 KEYWORD nonn,easy,nice,frac AUTHOR EXTENSIONS Extended and corrected by David Sloan, Sep 23 2002 STATUS approved

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Last modified October 21 22:47 EDT 2019. Contains 328315 sequences. (Running on oeis4.)