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%I M4456 N1886 #72 Aug 23 2022 08:49:28
%S 1,0,1,7,106,113,33102,33215,66317,99532,265381,364913,1360120,
%T 1725033,25510582,52746197,78256779,131002976,340262731,811528438,
%U 1963319607,4738167652,6701487259,567663097408,1142027682075,1709690779483,2851718461558,44485467702853
%N Apart from two leading terms (which are present by convention), denominators of convergents to Pi (A002485 and A046947 give numerators).
%C Disregarding first two terms, integer diameters of circles beginning with 1 and a(n+1) is the smallest integer diameter with corresponding circumference nearer an integer than is the circumference of the circle with diameter a(n). See PARI program. - _Rick L. Shepherd_, Oct 06 2007
%C a(n+1) = numerator of fraction obtained from truncated continued fraction expansion of 1/Pi to n terms. - _Artur Jasinski_, Mar 25 2008
%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 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="/A002486/b002486.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 P. Finsler, <a href="http://dx.doi.org/10.5169/seals-12812">Über die Faktorenzerlegung natuerlicher Zahlen</a>, Elemente der Mathematik, 2 (1947), 1-11, see p. 7.
%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 G. P. Michon, <a href="http://www.numericana.com/answer/fractions.htm">Continued Fractions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Pi.html">Pi.</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PiContinuedFraction.html">Pi Continued Fraction</a>
%H Eric Weisstein's World of Mathematics, <a href="http://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 3, 22/7, 333/106, 355/113, 103993/33102, ...
%p Digits := 60: E := Pi; convert(evalf(E),confrac,50,'cvgts'): cvgts;
%p with(numtheory):cf := cfrac (Pi,100): seq(nthdenom (cf,i), i=-2..28 ); # _Zerinvary Lajos_, Feb 07 2007
%t b = {1}; Do[c = Numerator[FromContinuedFraction[ContinuedFraction[1/Pi, n]]]; AppendTo[b, c], {n, 1, 30}]; b (* _Artur Jasinski_, Mar 25 2008; edited by _Harvey P. Dale_, Sep 13 2013 *)
%t Join[{1,0},Denominator[Convergents[Pi,30]]] (* _Harvey P. Dale_, Sep 13 2013 *)
%o (PARI) /* Program calculates a(n) (slowly) without continued fraction function */ {c=frac(Pi); print1("1, 0, 1, "); for(diam=2,500000000, cm=diam*Pi;cmin=min(cm-floor(cm),ceil(cm)-cm);\ if(cmin<c,print1(diam,", ");c=cmin))} /* or could use cmin=min(frac(cm),1-frac(cm)) above */ /* _Rick L. Shepherd_, Oct 06 2007 */
%o (PARI) for(i=1,#cf=contfrac(Pi),print1(contfracpnqn(vecextract(cf,2^i-1))[2,2]",")) \\ _M. F. Hasler_, Apr 01 2013
%Y Cf. A002485 (numerators), A072398/A072399, A063674/A063673, A132049/A132050.
%K nonn,easy,nice,frac
%O 0,4
%A _N. J. A. Sloane_
%E Extended and corrected by David Sloan, Sep 23 2002
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