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Search: id:A002486
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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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+0
15
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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; listen)
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OFFSET
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0,4
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COMMENT
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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 (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
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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.
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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.
G. P. Michon, Final Answers
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
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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*) - Artur Jasinski (grafix(AT)csl.pl), Mar 25 2008
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PROGRAM
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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 (rshepherd2(AT)hotmail.com), Oct 06 2007
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CROSSREFS
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Cf. A002485, A072398/A072399.
Adjacent sequences: A002483 A002484 A002485 this_sequence A002487 A002488 A002489
Sequence in context: A102461 A096131 A049210 this_sequence A075021 A138963 A002687
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KEYWORD
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nonn,easy,nice,frac
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AUTHOR
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njas
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EXTENSIONS
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Extended and corrected by David Sloan, Sep 23, 2002.
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