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 A002486 Apart from two leading terms (which are present by convention), denominators of convergents to Pi (A002485 and A046947 give numerators). (Formerly M4456 N1886) 21

%I M4456 N1886

%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 E. B. Burger, Diophantine Olympics ..., Amer. Math. Monthly, 107 (Nov. 2000), 822-829.

%D CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 88.

%D P. Finsler, Ueber 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 T. D. Noe, <a href="/A002486/b002486.txt">Table of n, a(n) for n=0..201</a>

%H Marc Daumas, <a href="http://www.ipsl.jussieu.fr/~omamce/SP/Oct00/Marc_Daumas.pdf">Des implantations differentes ...</a>, see p. 8.

%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, 2011. 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 (zerinvarylajos(AT)yahoo.com), Feb 07 2007

%t b = {}; Do[c = Numerator[FromContinuedFraction[ContinuedFraction[1/Pi, n]]]; AppendTo[b, c], {n, 1, 20}]; b (* _Artur Jasinski_, Mar 25 2008 *)

%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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Last modified May 21 01:52 EDT 2013. Contains 225472 sequences.