OFFSET
0,4
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, 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
REFERENCES
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).
LINKS
Daniel Mondot, Table of n, a(n) for n = 0..1947 (terms 0..201 from T. D. Noe, terms 202..1000 from G. C. Greubel).
E. B. Burger, Diophantine Olympics and World Champions: Polynomials and Primes Down Under, Amer. Math. Monthly, 107 (Nov. 2000), 822-829.
Marc Daumas, Des implantations differentes ..., see p. 8. [Broken link]
P. Finsler, Über die Faktorenzerlegung natuerlicher Zahlen, Elemente der Mathematik, 2 (1947), 1-11, see p. 7.
Henryk Fuks, Adam Adamandy Kochanski's approximations of Pi: reconstruction of the algorithm, arXiv preprint arXiv:1111.1739 [math.HO], 2011-2014. 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
EXAMPLE
The convergents are 3, 22/7, 333/106, 355/113, 103993/33102, ...
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, Feb 07 2007
MATHEMATICA
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 *)
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, 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
CROSSREFS
KEYWORD
nonn,easy,nice,frac
AUTHOR
EXTENSIONS
Extended and corrected by David Sloan, Sep 23 2002
STATUS
approved