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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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36
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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).
CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 88.
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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Eric Weisstein's World of Mathematics, 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, Feb 07 2007
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MATHEMATICA
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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 *)
Join[{1, 0}, Denominator[Convergents[Pi, 30]]] (* Harvey P. Dale, Sep 13 2013 *)
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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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KEYWORD
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nonn,easy,nice,frac
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AUTHOR
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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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