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A002486
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Apart from two leading terms (which are present by convention), denominators of convergents to Pi (A002485 and A046947 give numerators).
(Formerly M4456 N1886)
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21
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1, 0, 1, 7, 106, 113, 33102, 33215, 66317, 99532, 265381, 364913, 1360120, 1725033, 25510582, 52746197, 78256779, 131002976, 340262731, 811528438, 1963319607, 4738167652, 6701487259, 567663097408, 1142027682075, 1709690779483, 2851718461558, 44485467702853
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OFFSET
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0,4
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COMMENTS
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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
a(n+1) = numerator of fraction obtained from truncated continued fraction expansion of 1/Pi to n terms. - Artur Jasinski, Mar 25 2008
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REFERENCES
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P. Beckmann, A History of Pi. Golem Press, Boulder, CO, 2nd ed., 1971, p. 171 (but beware errors).
E. B. Burger, Diophantine Olympics ..., Amer. Math. Monthly, 107 (Nov. 2000), 822-829.
CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 88.
P. Finsler, Ueber 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).
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..201
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
Eric Weisstein's World of Mathematics, Pi.
Eric Weisstein's World of Mathematics, Pi Continued Fraction
Eric Weisstein's World of Mathematics, Pi Approximations
Index entries for sequences related to the number Pi
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EXAMPLE
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The convergents are 3, 22/7, 333/106, 355/113, 103993/33102, ...
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MAPLE
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Digits := 60: E := Pi; convert(evalf(E), confrac, 50, 'cvgts'): cvgts;
with(numtheory):cf := cfrac (Pi, 100): seq(nthdenom (cf, i), i=-2..28 ); # Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 07 2007
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MATHEMATICA
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b = {}; Do[c = Numerator[FromContinuedFraction[ContinuedFraction[1/Pi, n]]]; AppendTo[b, c], {n, 1, 20}]; b (* Artur Jasinski, Mar 25 2008 *)
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PROG
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(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 */
(PARI) for(i=1, #cf=contfrac(Pi), print1(contfracpnqn(vecextract(cf, 2^i-1))[2, 2]", ")) \\ - M. F. Hasler, Apr 01 2013
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CROSSREFS
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Cf. A002485 (numerators), A072398/A072399, A063674/A063673, A132049/A132050.
Sequence in context: A049210 A167814 A209545 * A203971 A145167 A141358
Adjacent sequences: A002483 A002484 A002485 * A002487 A002488 A002489
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KEYWORD
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nonn,easy,nice,frac
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Extended and corrected by David Sloan, Sep 23, 2002.
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STATUS
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approved
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