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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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20
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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, 136308121570117, 1816491048114374, 1952799169684491
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| 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 (rshepherd2(AT)hotmail.com), Oct 06 2007
a(n+1) = numerator of fraction obtained from truncated continued fraction expansion of 1/Pi to n terms. - Artur Jasinski (grafix(AT)csl.pl), Mar 25 2008
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REFERENCES
| 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
| T. D. Noe, Table of n, a(n) for n=0..201
Marc Daumas, Des implantations differentes ..., see p. 8.
G. P. Michon, Continued Fractions
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Eric Weisstein's World of Mathematics, Pi Approximations
Index entries for sequences related to the number Pi
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EXAMPLE
| The convergents are 3, 22/7, 333/106, 355/113, 103993/33102, ...
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MAPLE
| 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
| b = {}; Do[c = Numerator[FromContinuedFraction[ContinuedFraction[1/Pi, n]]]; AppendTo[b, c], {n, 1, 20}]; b (* Artur Jasinski (grafix(AT)csl.pl), Mar 25 2008 *)
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PROG
| (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 (rshepherd2(AT)hotmail.com), Oct 06 2007 */
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CROSSREFS
| Cf. A002485, A072398/A072399.
Sequence in context: A096131 A049210 A167814 * A203971 A145167 A141358
Adjacent sequences: A002483 A002484 A002485 * A002487 A002488 A002489
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KEYWORD
| nonn,easy,nice,frac
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Extended and corrected by David Sloan, Sep 23, 2002.
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