A002486 Apart from two leading terms (which are present by convention), denominators of convergents to Pi (A002485 and A046947 give numerators).

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice
• frac

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

Examples

			The convergents are 3, 22/7, 333/106, 355/113, 103993/33102, ...
		

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).
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 274.

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 différentes ..., 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
• Index entries for sequences related to the number Pi

Crossrefs

Cf. A002485 (numerators), A072398/A072399, A063674/A063673, A132049/A132050.

Programs

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

Join[{1,0},Denominator[Convergents[Pi,30]]] (* Harvey P. Dale, Sep 13 2013 *)

PARI

for(i=1,#cf=contfrac(Pi),print1(contfracpnqn(vecextract(cf,2^i-1))[2,2]",")) \\ M. F. Hasler, Apr 01 2013

Extensions

Extended and corrected by David Sloan, Sep 23 2002